Good morning, everyone, and welcome to the first meeting of the Social Justice and Social Security Committee. It seems like such a long time ago now, but seeing as this is the first time that we've met, it seems only right to say happy new year to everyone. Our first item of business today is a decision to take item 4 in private. Are we all agreed? Agreed, thank you. Our next item is an evidence session on the Scottish Government's budget 23-24, which was published on 15 December. I welcome to the meeting, Shona Robison, Cabinet Secretary for Social Justice, Housing and Local Government. Joining the Cabinet Secretary today, we have Shirley Lang, director for tackling child poverty and social justice, and Kevin Stevens, head of strategic and programme finance at the Scottish Government, and all are joining us in person today. Before we move to questions, I would invite the Cabinet Secretary to make an opening statement. Thank you, convener, for inviting us to the committee today. We have again faced a very challenging fiscal context in which to set this budget, and we are all aware of the impact on our economy, of the UK Government's mini-budget and the effect of rising prices as a result of Russia's invasion of Ukraine. We are therefore pleased that we have continued to prioritise our resources towards taking steps to tackle child poverty, alongside our other priorities of protecting public services and investing in the transition to net zero. Within my portfolio in 2023-24, we are committing £5.2 billion in social security and welfare payments, which will go directly to over 1 million people in Scotland. That, of course, is money that will help low-income families through the cost of living crisis, support older people to heat their homes in winter and enable disabled people to live full and independent lives. That includes investing £442 million in the Scottish child payment, the UK's most ambitious child poverty reduction measure, which we increased to £25 per week per child in November 2022, when we extended the payment to all eligible six to 15-year-olds. In 2023-24, around 387,000 children are estimated to be eligible. This investment underlines our commitment to delivering on our national mission to tackle child poverty using all the powers and resources that are available to us to support families as far as possible and tackle the underlying causes of inequality. It sets out how we will deliver on the priority action set out in the best upright futures published in March 2022 with a key focus on long-term parental employment support, increased social security and measures to reduce household costs. Across social security, our total investment is over £770 million above the level of funding forecast to be received from the UK Government through the block grant adjustments. Money that, as I said earlier, will go directly to the people of Scotland who need it most. We upgraded Scottish child payment in November 2022 by 25 per cent when we extended it to six to 15-year-olds and we will rate all other Scottish benefits in line with inflation by 10.1 per cent in April 2023 at a cost of around £430 million. That rating includes benefits where there is a statutory requirement as well as those where uprating is discretionary in recognition of the difficulties being faced by many due to the increased cost of living. We are investing over £520 million to deliver devolved social security in Scotland in 2023-24 ensuring a simplified, compassionate system that will treat everyone with dignity, fairness and respect and provide people with an improved experience. In February 2023, subject to DWP providing the necessary data, we will launch our new winter heating payment with a stable, guaranteed annual payment of £50, which will help around £415,000 low-income individuals with their heating expenses each winter. We are also investing £41 million, including local authority administration in the Scottish Welfare Fund, which, while delivering an independent review, provides essential help to the most vulnerable in our communities. The budget also recognises the important role carers play in supporting people with disabilities or long-term conditions. We will invest more than £420 million through the carers allowance, the carers allowance supplement and the young carer grant. Alongside that, we will begin to roll out the Scottish carers assistance by the end of 2023 with full national introduction in the spring of 2024, delivering an improved service designed with carers to meet their needs and in line with our principles of dignity, fairness and respect. Within the budget, we are also making available £752 million for affordable housing as part of the wider planned investment of over £3.5 billion over this Parliament to deliver more affordable and social homes. We are making available £85.9 million for discretionary housing payments to support tenants who are struggling with their housing costs. Over £12 million to support continued progress of actions in our Ending Homelessness Together Action Plan, which focuses on homelessness prevention, rapid rehousing and housing first. Finally, we are investing over £70 million in our Ukrainian resettlement programme to ensure that those displaced by the illegal war in Ukraine continue to receive a warm Scots welcome and are supported to rebuild their lives in our communities for as long as they need to call Scotland their home. In conclusion, against a backdrop of a challenging economic backdrop and an on-going cost crisis, this budget and my portfolio continues to seek to prioritise action in our three main areas, including tackling child poverty. I thank the committee for its pre-budget scrutiny. I look forward to your questions. Thank you very much Cabinet Secretary. You mentioned in your statement some of the difficulties this budget is set against this year. Can you perhaps expand or outline what impact the cost of living crisis and the inflationary pressures that have had on this budget and how this is different from previous budgets? The cost of living crisis is huge. We have therefore tried to prioritise support to households, particularly low-income households. Over the past year, we invested £3 billion, a billion of which was only available in Scotland. Some of that was money that had to be reallocated, that had to be re-prioritised to ensure that we were able to do that. Looking forward, the levels of pressure on household budgets remain the same, so we are looking to do what we can through this budget to support household incomes. We do that against the backdrop of a very difficult financial settlement in terms of the autumn statement and the consequentials. We will be aware of the reductions in capital allocations. Over a number of years, the pressures on capital budgets and resource budgets added to that, the levels of inflation mean that every pound we invest is worth less than it was last year and particularly the year before that. That is before we consider the impact of the pandemic, as it continues to impact on the recovery of public services. All of that is a number of backdrops that no-one would want to try to move forward to support households to tackle child poverty. It is an extremely difficult backdrop, but we have tried to prioritise support to low-income households as best we can. It is extremely difficult indeed. Thank you, Cabinet Secretary. I will move to questions from members now. First, I have questions from Pam Duncan-Glancy, who is joining us online. Thank you, convener, and good morning to Cabinet Secretary and your officials. In the Government's response to the committee's pre-budget report, they recognise the employability support plays an important role in tackling child poverty. We are conducting an inquiry as part of that just now. Can the cabinet secretary set out what she believes the consequences of the reduction in funding to employability support this year will have been? First, let me be clear that the draft budget sets out our plans to invest £69.7 million to support the scaling up of employability support to parents in 2023-24. That represents a reinstatement of the in-year budget reduction announced by the Deputy First Minister in the emergency budget review. Although the decision to take £53 million as a saving in 2022-23 was not taken lightly and the Deputy First Minister laid out the incredibly difficult backdrop to having to make that emergency budget review but it was taken in recognition that employability interventions take a bit longer to have an impact and were unlikely to result in immediate increases in income for people. While the decision has meant that services were not able to begin scaling up support in 2022-23, as we would have liked, the impact of employability was not factored into the modelling of the interim targets, for example, in recognition of the fact that there would be a time lag between someone entering support and sustaining employment. It was also noted in the published annex that the impact of employability has set out in the plan was quite optimistic with outcomes potentially being achieved later and at a slower rate. I guess a lot of that depends on the labour market and some of the other backdrops that we are seeing in terms of the pressures and some of the challenges in terms of reserved benefits of people moving in to work. However, as I started off saying, the reinstatement of that resource means now that employability work can scale up as was anticipated in the plan. Can the cabinet secretary set out what percentage of outstanding child, Scottish child payment applications were processed by Christmas and is there still a backlog for applications? Are you aware of any? I will hand over to Kevin in a moment. It is fair to say that there has been a huge response to Scottish child payment eligibility criteria being widened and clearly that is a good thing, but it has meant that the staff at Social Security Scotland have been working extremely hard to make sure that people's payments are processed as quickly as they possibly can be. I would want to pay tribute to the efforts that they have made. Kevin, I do not know if you have any further figures that you could say in terms of the processing of Scottish child payment, or is that something that needs to come back to the committee on? I think that that is something that we should come back to the committee on. What we will do, if that is okay, is to come back with the latest figures, if that is okay. That would be very helpful, cabinet secretary. Pam, do you have further questions? Yes, just a short response to that and then one further question. Thank you, cabinet secretary. That would be appreciated. I understand that there was a significant response to that, which I do not think that we can really be surprised at, given that the group of people who were applying for the payment at that point had been entitled to it for a number of years, but had not been able to access it. I am not surprised by the scale of interest, and I would hope that neither was cabinet secretary nor Social Security Scotland, and that that was impacted into the announcement in November to do so, to roll that out and it was planned properly.  First of all, on your latter point, there was a scaling up of staff and expectation that that number of applications would increase, and therefore there was additional gearing up towards that. As I said, we will get those figures to the committee. On the targets, first of all, we will go back to the cumulative impact assessment that you published in March, as you know, projected that around 17 per cent of children will live in relative poverty in 2324, based on the impacts that will be made, not least by the Scottish child payment that we have just been discussing. We have done absolutely everything we can to put into the best chance of hitting those targets, and we have done everything we can in a very difficult situation. What I would say is that the current economic climate, much of which is outwith our control, the impact of the cost of living, the impact on household budgets, the challenges that people who are in work who are not even entitled to Scottish child payment are facing at the moment, is not the most ideal backdrop to be trying to reduce levels of child poverty, but despite that, we are doing absolutely everything we can, and we remain firmly committed to delivering them. We will not know for sure whether the targets have been met, as you know, until official statistics are available in March 2025. As I said, the plan is the right plan, it remains the right plan, but as Pam Duncan-Glancy and others will be aware, what the plan cannot do is to take account of economic circumstances in the UK, and decisions that are made by the UK Government. I could take just one, and that is the freezing of the local housing allowance rates, yet again at 2020 levels, so this is for a third-year running, is another real-terms cut, and my fear is that it will exacerbate the situation for people in Scotland who are renting.  That is one example, there are many others, but that is one concrete example of where decisions made elsewhere have an absolute negative impact on the plans that we have in place. Pam Duncan-Glancy, I will move now to questions from Jeremy Balfour. Good morning, cabinet secretary, and good morning to your officials. I wonder if I can just, on the Scottish welfare fund, you know that we have been quite a lot of discussions about that over the years in regard to how it is delivered and how local authorities delivered. Just before the post-pandemic, and as we come out of this, have there been any thinking in regard of the Scottish Government of how we deliver that benefit, whether the way that we do it through local authorities is the best way to get it targeted to the most people? Is there any thinking of reviewing how we deliver the payment? As you know, it is being reviewed, and we did not set any barriers or restrictions in terms of that review and what recommendations would come back for change, whether that is on how the welfare fund is delivered. You will be aware that some of the issues that sparked that review were around inconsistency, that in some local authorities it was underspent and some it was overspent. All of those issues are being looked at as part of the review. I think that we probably should wait until that review is concluded. I am very happy to come back to the committee once we see what the recommendations are. The reason that local government is the distributor of the welfare fund was very much with the view in mind of knowing local needs, being able to respond to the needs of local people who were requiring crisis support. I think that that would be more difficult to do from a national agency, for example. That level of discretion of its use. As I say, we probably need to wait for the outcome of the review, which is this spring, if I am correct. As I say, I am happy to come back to the committee once we have that. Just before Christmas, we took evidence from David Wallace from Social Security Scotland. One of the issues that came up was the processing time. Within the charter, it is a 10-day processing time measure. I think that his words were incredibly challenging. I think that it was clear that he felt that going forward that was not going to be met on a consistent basis. I wonder whether you are going to look at that time, because, presumably, there is no point in having a time that no one is ever going to hit. A couple of things about that. The commitment in the charter is to handle the client's application and enquiries as quickly as we can, and that is Social Security Scotland's commitment. That is still the case. Alongside nearly 70 other performance measures, the charter measurement framework considers what proportion of applications for low-income benefits were processed within 10 days. The background to that measure was chosen at a time when the framework was being developed back in 2019. I think that it reflected the 10-day statutory timeline for processing funeral support payment applications specifically around that time. That is what it was based around. That was before the introduction of many of the current benefits. It is a very different organisation than it was back in 2019, including the Scottish child payment that we have just been talking about in terms of the number of applications that have come in. I guess that taking together with the other measures, the measure helps to provide context on the performance against our charter. I do not think that it was ever a target as such, and it was not intended to reflect standard processing times across all low-income benefits. It is not that it is not important, but what is probably most important is getting it right. Jeremy Balfour and other members of the committee will be aware of some of the complexities around some of the disability payments, for example, of making sure that the information is gathered. Sometimes that information is coming from various sources and to get it right first time rather than having to revisit decisions. While processing the measurement rather than the target is important, it has to be seen in the context of all the other benefits that Social Security Scotland is now administering. I do not know, Kevin, if you have anything to add to that. Other than to say that the charter does not contain a commitment to processing applications within 10 days, and it is not a target for social security Scotland. However, the charter measurement framework, which is a list of measures relating to the commitments in the charter co-designed with people with lived experience of social security, includes a measure of what percentage of applications for low-income benefits were processed within 10 working days. The measure dates from 2019 when social security was only delivering seven benefits. It is one of almost 70 measures that Social Security Scotland collects data on and which, taken together, give the context on the agency's performance against the promises in their charter. In summary, it is important, but it is not the only measurement. It is something that we should keep an eye on. If you follow Social Security Scotland on Twitter, you will see that many people are saying that the targets have been very bad. There are long delays. If we look at the number of people who are making, having to do a reconsideration, the number is growing, so we are not getting it right first time. Do you think that we need to set realistic targets so that the public has a genuine understanding of what we can expect from Social Security Scotland? At the moment, I think that the perception is that, in many cases, they are failing to get that decision on time or right. I think that any agency could never say that it is going to get absolutely every decision right first time within a certain timescale. But what we can say about Social Security Scotland is that the customer satisfaction rates are very, very high indeed. I could say that that is in marked contrast to other agencies, the DWP in particular. The point about reconsideration is an important one. That is not to say that there won't have to be some considerations, but trying to make sure that, where possible, they get the decision right first time. That is about making sure that all the information is there. That sometimes takes time and that sometimes takes time from external agencies. It is not all within the gift of the staff within Social Security Scotland. When you consider the volume of applications, in a quite a short space of time, in the growth of benefits and supports that Social Security Scotland is now responsible for, it is a huge growth in the organisation. I think that we need to see in context the fact that, for some people, it would have wanted the decision sooner and for some people they are not happy but there has to be a reconsideration. For the overwhelming majority of customers of Social Security Scotland, there has been a very good service and customer satisfaction rates are very, very high indeed. It shouldn't be complacent, no? We should always look at those things, we should always get that feedback loop from those with lived experience and, again, Social Security Scotland is very good at that, of that regular checking with those who receive support to make sure that, if there are further improvements that can be made, then those are made. I have a final question on this area for me. I might add in regard to the charter, as you will be aware, I think that we need January next year and, obviously, there is a process that you will go through in regard to that with consultation. I am just wondering at this point do you know when any alterations or any changes to that charter will be able to be scrutinised by the committee before they are laid, before Parliament? Will you give a guarantee that there will be an opportunity for the committee to take evidence around that in time before we are laid? If the committee wants to do that, of course that is for the committee to do and what we can do is come back to the committee with some timeframes and if that would be helpful just so you know for your own planning. I think that that would be helpful and I suspect to know the answer to this but is there any thought within Government because there was quite a long debate when the bill was going through Parliament or whether the charter should have legal authority? We decided against that as a Parliament. Is that going to be part of the review to look at that to see whether it should be given some statutory context? No, I do not think that that is the focus but again we can come back to the committee with more detail around the timeframe and the scope. Thank you cabinet secretary. Thank you very much Jeremy. We will move back to Pam Onley. Thank you for that. I was wondering probably to follow on a little bit from what my colleague Jeremy Balfour was asking. David Wallace of course did say in the committee last week that it was always going to be a challenge to hit an attendee figure. It feels like having that would be meaningless if we are not going to meet it and I wonder what further work the cabinet secretary can do to help us get to that position. As I said and Kevin Stevens has said it is not a target as such. It was more a measurement alongside 69 others around performance and I guess with the charter being reviewed in the committee's opportunity to look at that and as I said we will get you the details of timelines and scope about whether or not the measurements are still fit for purpose. I guess when you look at the organisation's growth when this was set back in 2019 it was a very different organisation so perhaps part of the look at the charter is to reflect the fact that it is now a much, much larger organisation dealing with and processing applications from a far greater group of people and processing a larger number of supports and benefits. I think that the context is very different now and perhaps that is something that can be looked at as part of the review of the charter. If I can just pick up on something that the Scottish Fiscal Commission said which is kind of in a similar vein about administration and budgets and search obviously. They said last week that the data that they were getting to make the assumptions about the financial forecast going forward some of which was really kind of it wasn't published they were getting it almost through the back door I guess. Is there a commitment from Government to encourage social security Scotland to be publishing the data that is used for the Scottish Fiscal Commission to make its forecast so that everyone can see it transparently? I'll pass on to Kevin in a moment. I mean, I don't think there's any attempts to not provide or be opaque in any way with the Scottish Fiscal Commission. It wouldn't be in our interests or social security Scotland's interests to do so given the importance of the forecasts that the Fiscal Commission make and they make adjustments based on that so it's in our interests to make sure they get that right and it's accurate. Kevin, I don't know if there's anything you want to add in terms of the speed at which that information is provided. I mean, we are both agency colleagues, statisticians analysts are in regular contact with the Scottish Fiscal Commission all the time. We work closely together with excellent lines of communication and we're working to give them the information that they need because it is important for forecasts. Thank you, I appreciate that. I don't doubt that there are strong lines of communication that would just be really helpful if those lines were more in the public domain, I guess. If it's possible to have a commitment to encourage social security to do that then that would be helpful. Are you able to do that kind of searching? I think that we try to be as transparent with information and figures as we can and it's certainly something we can have a look at if there's more we can do in that space. Thank you, that would be helpful. I don't have any further questions at this point. Thank you very much Pam. Continuing this theme, we'll move to James Dornan and officials. Can I ask the cabinet secretary almost all the budget growth in this portfolio is from the £1.2 billion 30% increase to the forecast spend and social security benefits due mainly to PIP, stroke, ADP, the Scottish child payment and increasing benefits in line with inflation. Can you tell me a bit more about why the Scottish Government made these choices why they decided to increase benefits in line with inflation and increase Scottish child payment by 150% and why the Scottish Government spending more on ADP compared to what it was doing under PIP? I think that it goes back to the points I made at the start in my opening remarks that we see this as a priority to tackle child poverty and poverty among some of the most marginalised and low-income households in our society. Even before the cost of living crisis the fact that we have established our own social security system in Scotland is a reflection of the fact that we wanted to do more to support people and to have a different system based on dignity respect and fairness and the funding and budgetary decisions really follows that commitment so whether that's on the Scottish child payment or indeed the investments that we are making in the other benefits reflects that the largest part, the £208 million of the difference on personal independence payment where the SFC forecast an increase in the number of successful new applications and partly due to the prevalence of mental health conditions worsening long-term health conditions and the cost of living crisis is leading to people more likely to apply so I think in recognition of that that is why the forecasts are as they are and I guess the other point to make is that the £776 million above the level of funding from the block grant adjustments of course is money that we are having by and large to find out of a fixed budget and I guess that comes back then to some of the key issues around the fiscal framework and the fact that that needs to be reviewed which it's been some progress made around but still very much at the foothills of those discussions with the UK Government and borrowing powers I mean you look at the actions taken by most other countries during the cost of living crisis they are able to use borrowing powers in order to respond to a cost of living crisis whereas we've had to find additional money within a fixed budget and that has its limitations but those are decisions that we've made because we believe that they are the right decisions and the right priorities Yes, thank you Just on that last point of the borrowing powers I noticed that Dr Hoase of the Scottish Human Rights Council indeed with you about that being a significant limitation but I'd like to ask you about the impact on UK welfare policies Ed Pibus of the children poverty action group talked about the two child limit said larger families are much more likely being poverty because of the UK welfare system capping the amount of support that can get to only two children and of course that's going to drive larger families into poverty a lot of lone parents of course are also young parents and young parents are hit by the under 25 penalty in university credit on to those points from the CPAG We agree with the child poverty action group and of course we've written to the UK Government on several occasions calling for the two child limit and the benefit cap to be scrapped Scottish Government is already mitigating the benefit cap through local authorities however we acknowledge that this would be more effective if it was done at source by the UK Government and as a Government we also agree that universal credit should be paid at the same amount no matter the age of the person applying and this of course would help many people and families who are facing hardship due to what is quite an arbitrary age discrimination that the UK Government has introduced I think that is in sharp contrast to the significant level of support that the Scottish Government is providing to low income to low income parents so we have a lot of sympathy with what the child poverty action group is saying Yeah, as Mr Pibus said you may get less money being under 25 but the energy bills are exactly the same Can I ask you is there any analysis that the Scottish Government carried out about the impact of the UK Government welfare policies on the levels in Scotland? Well there has been a lot of independent analysis made over the years and I think one of the most important was around the last welfare reform report published in April last year estimated that reversing key UK Government welfare reforms that have occurred since 2015 could bring an estimated 70,000 people out of poverty in Scotland including 30,000 children in 2023-24 and that of course would very much help with the meeting of the child poverty targets Okay, thank you very much Government Secretary no further questions, convener We'll move now to questions from deputy convener Emma Roddick I'm just looking at the Scottish Government's spending on discretionary housing payments which is a significant figure is the cabinet secretary able to share any data on how many people have been saved from spending money on the bedroom tax here in Scotland and how many will be supported by extending these payments to mitigate the benefit cap So the £85.9 million investment I think in discretionary housing payments provides absolutely critical support. The budget has increased from 2022-23 because of the funding added to mitigate the benefit cap as fully as we can within devolved powers and anyone in receipt of housing benefit or universal credit with a housing element will be able to claim a DHP towards the amount that their benefits have been reduced by the cap So in 2022-24 we're going to make £83.7 million available to local authorities to spend on DHPs and £69.7 million to fully mitigate the bedroom tax is going to help over 91,000 households in Scotland to sustain their tenancies and we will invest £6.2 million from April to mitigate the benefit cap and £7.9 million to mitigate against the impact of other UK Government welfare cuts including the on-going freeze to local housing allowance rates referred to earlier on which has a major impact on renters given that it's been frozen for the third year in a row So these are important investments but again they're out of a fixed budget so we would be able to spend that money on something else On that point discretionary housing payment is obviously just one example of many ways in which the Scottish Government is mitigating decisions elsewhere Could the cabinet secretary expand on any targeted action that the Scottish Government has asked the UK Government to take to help tackle the cost of living crisis? We have written to them over a number of times asking them to use their levers The First Minister herself actually wrote to the Prime Minister towards the end of last year and it includes things like further targeted financial support to low-income households particularly around energy energy costs for social security benefits to increase with inflation and a permanent £25 uplift to universal credit that obviously we saw during the pandemic but then was removed The reversal of the two-child limit for UC and tax credits the abolition of the benefit cap We thought they could go further on the windfall tax and we have gone some way on that but we think they could have gone further on the windfall tax additional funding obviously for devolved Governments not just our own but others as well to help to provide support to people so we have made those representations on a number of occasions Thank you Moving on to theme 3 which is homelessness we'll start off with questions from Miles Briggs Good morning cabinet secretary good morning to your officials as well The submission to the committee shelter said that freezing funding for homelessness services and cutting funding for the delivery of new social homes is not in line with the Scottish Government's international obligations to progressive realisation of rights How would you respond to that? So let me first of all take the issue of the affordable housing supply programme budget now it was to had a bit of an exchange this at the committee on Tuesday so I hope Miles Briggs won't mind me just going over some of the key points on this again so first of all let me be clear that the £3.5 billion programme on affordable housing remains £3.5 billion there's no change to that but the profiling of it was always going to ebb and flow depending on the availability of resources and I would just point out that obviously our capital resources available to us are very much constrained by the UK Government decisions so the capital availability remain at the same cash level and we've seen a 3.4 percent real reduction in our UK Government capital allocation between 2022-23 and 2023-24 so context is important if you add into that the impact of high inflation that means that every pound available for the affordable housing supply programme in this coming year is worth less than it was last year and much less than it was the year before so context is important but the £3.5 billion remains the same now in terms of the funding for this coming year there is a £37 million reduction on what was previously published which is 4.7 percent that was previously published capital spending review figure however we are taking steps to mitigate that because we know the importance of this programme and there are three ways we're doing that first is on financial transactions so we've got additional financial transactions available to the value of £17.6 million we also have a £15 million transfer from the net zero budget to help with net zero heating systems within the affordable housing supply programme and thirdly we will have additional money from the charitable bonds so taking all of that together the gap will be reduced to a very very small figure if anything at all the main problem here is the issue of inflation and I met with Shelter and other colleagues at the strategic housing group meeting homelessness group meeting yesterday and they all said the same that the major problem here is inflation and the value of that money SFHA said exactly the same that the ability to deliver projects on budget are really constrained because of inflation and interest rates in terms of homelessness over £12 million included in the budget to support continued progressive actions in our ending homelessness together action plan on top of the homelessness funding provided through the local government settlement we're investing £100 million up to 2026-27 to continue to transform the homelessness system that's in addition to a budget of £30.5 million which is actually an increase to local authorities for homelessness prevention activity in 2023-24 and that funding is going to be distributed using a revised formula that really better represents the drivers of need in homelessness and recognises local authority efforts to reduce the use of temporary accommodation and Miles Briggs will be aware of the work that is going on with the task and finish group of which Shelter are a co-chair in order to give me some recommendations around what needs to change but I think what is very clear that the prioritisation of funding and particularly grants available to the third sector need to pivot towards two clear priorities and reducing people in temporary accommodation and secondly, prevention of homelessness so I think you'll see a far sharper focus on both of those priorities with the funding that's going in Thank you for that I wanted to ask specifically and I've raised this consistently with the cabinet secretary around temporary accommodation if you'd like to point out that those recommendations are due shortly that's likely to be after this budget is passed in Parliament Will additional funding then be made available specifically for that and any policy changes which I hope will be brought forward? So in 2020-24 as I've just said we're giving all local authorities an annual share of £30.5 million for homelessness prevention so that and an annual share of £8 million to support the rapid rehousing transition plans we will look at how we make sure that the priorities and recommendations emerging from that group are reflected in the spend now I obviously need to see what those recommendations are some of them may be around resourcing but some of them are probably going to be more around the prioritisation and how we use the existing money in the system to focus more on the issues of temporary accommodation and indeed the issues of prevention. I've also been meeting with housing conveners to look at what more can be done particularly in those areas where there are the biggest problems and I'll come on to that in a second but I've been encouraging proposals to look at bringing empty properties back into use and also looking to be more strategic so the £50 million Ukraine long-term resettlement fund I've been encouraging local authorities to bid into that with a view to long-term investment so not just for Ukraine displaced persons but actually bringing properties back that can be there for long-term use and I think that is important with specific reference to Edinburgh and Glasgow where clearly the vast bulk of the issues of temporary accommodation lie. I have asked both Edinburgh and Glasgow to come forward with proposals and what I've said is that I would look very favourably upon any specific proposals that are going, if they can demonstrate that it's going to make an impact on the use of temporary accommodation I will look very favourably on such proposals and I understand that Edinburgh are working on something, we've not seen the detail of that and of course they did receive some extra money last year for the affordable housing supply programme but I have made that offer and we'll see what comes back. Thank you for that. I'm not going to rehearse the arguments from local government committee about funding which COSLA have highlighted their concerns around but I think there is concern over where potentially homelessness could be lost in translation and services funded through joint services. The government is bringing forward homelessness prevention legislation and also homelessness specific funding. I wondered in terms of making sure that that funding at a local level is going towards delivering on these priorities, how the cabinet secretary is going to achieve that. We had quite a long discussion about this Tuesday's committee meeting around that balance of the demand from local government to more flexibility which I'm sympathetic to but with the need to make sure that discrete areas of spend like homelessness spend delivers on the progress that we need to see deliver. If you look at the progress that's being made on housing first and the rapid rehousing transition plans I guess the question for us collectively is would that have happened if that money hadn't been earmarked for that purpose and these are not easy things to balance with local government, we're sitting around this table now they would say well these are joint priorities so you need to trust us to get on and deliver that so we need to have assurance and we have joint accountability for the delivery of these joint priorities so these discussions will continue I am really keen to make sure that local government has the flexibility in a very challenging financial backdrop and again we discussed this at length around the £570 million that has been made available for local government and in a very very difficult financial backdrop bearing in mind that there were only £800 million of consequentials coming from the UK Government budget settlement so we will no doubt continue to have these debates but within that global amount there are some really important discrete areas of spend and I think this is definitely one of them that we would need assurance that the good work that has been done jointly with local government continues okay thank you thank you we'll now move on to questions from Faisal Chowdry who is joining us online thank you very much good morning I had a question on homelessness but the Cabinet Secretary had already answered on the temporary accommodations and homelessness can I go on theme 4 which is a question on refugees and asylum seekers absolutely Faisal carry on thank you very much what support can be offered to non-Ukrainian refugees and asylum seekers particularly those who have access to public funds due to their immigration status to ensure equal rights for all refugees and asylum seekers in Scotland well obviously this is an important question and as part of ending destitution together strategy published jointly with COSLA the Scottish Government is funding the Scottish Crisis Fund project delivered by the British Red Cross the project aims to support people across Scotland who are facing crisis situations and is delivered inclusively of people with no recourse to public funds it provides crisis funds via a cash distribution network and also helps with wider advice and support for people facing destitution understanding the level of demand for cash assistance is one of the key objectives of the project and the British Red Cross is collecting data on grants delivered through the project which will help to build a picture of the circumstances of people facing crisis situations including people with no recourse to public funds and I guess it should be noted that this data won't identify the number of people with no recourse to public funds in Scotland as a condition which is widely applied to people who are subject to immigration control under UK immigration law including international students and people on employment visas so it's very difficult to get those figures what I can also say is since April 2022 the Scottish Government has provided £362,500 to the British Red Cross to support the delivery of that Scottish Crisis Fund projects and officials are finalising funding for the rest of the financial year so we recognise that this is a really vulnerable group of people and there are huge limitations as I'm sure Foisal Troudery will know around the support that we've given to public services and we are very constrained by UK Government rules around that but we have, through the British Red Cross provided this very important crisis fund. Thank you very much I don't have any other questions on this. Thank you Foisal we'll move now to questions from Paul MacLennan Thank you, convener and good morning Cabinet Secretary in the panel I just want to move on to talking about fair work in the third sector I chaired across party group and social enterprises and one of the issues that's coming through is looking at multi-year funding but also to try and support the sector amongst, of course you mentioned inflation earlier on as well just to say a little bit more about what the Government's doing to support third sector organisations we really obviously recognise the impact that rising operating costs is having across society including on charities and voluntary organisations in the third sector and we do continue to invest widely in the third sector and have committed to increase multi-year funding wherever possible and we will do that and fairer funding principles prefer multi-year arrangements for third sector organisations where appropriate so I think we want to move to a position where multi-year grants are the default unless short-term funding which of course there are some projects that will receive one off one year funding but where that's not the case I would like to move to that being the default normal process and I'm really committed to progressing that within my portfolio all cost factors are taken into account when considering proposals for grants from third sector organisations and I guess it's for individual grant makers and funders to determine the value of individual grants but the real living wage conditionality is concerned with really uplifting low paid workers to at least the real living wage and that is an important part of the fair work principles and any further pay increases an employer wishes to introduce above for the employer concerned you know it is a challenge for third sector and I don't think we can shy away from that but it is important that we make progress and I think that the multi-year funding will at the very least help third sector organisations to plan beyond the kind of year to year okay thank you for that and I was going to ask about the fairer funding and the other issue I wanted to ask about and you can you touch on it on and you'll know I raised this on local government committee on Tuesday was talking about fiscal framework discussions that are going on you talked about additional borrowing and obviously this committee is probably more demand led than any other committee so I don't know if you want to say any more on or just on that again and Kenny comes back to the issues you talked about the three year funding and it's part of the discussions around the fiscal framework but also additional borrowing powers in terms of looking at who could have forecast inflation would have been 10, 11 per cent 18 months ago a year ago Scottish Government has to then find that funding at some stage and take it from elsewhere Indeed all of that absolutely true and the current fiscal framework arrangements clearly constrain the extent to which we're able to respond to this or any other crisis said something about that earlier on around the largely fixed budget constrain borrowing and reserve powers limit all of which is very difficult and you add to that inflation then it is extremely difficult so the fiscal framework review really has to ensure that the Scottish Government and this Parliament have the necessary fiscal flexibility to manage the risks that we face now and in the future within our devolved responsibilities and to support economic recovery and we have to really get a different set of arrangements out of that so it's due to be reviewed after the completion of an independent report that was jointly commissioned by the UK and Scottish Governments being reached with the Treasury counterparts on the detailed arrangements for the review so when I said we were at the foothills I guess that's what I was alluding to and the final version of the report has been submitted to both Governments for consideration but the timing and arrangements for publishing the report as I understand are still under discussion so we this is I couldn't emphasise how important this is it's really really important that we get this right and we I'm sure the committee will be keeping a watching brief on a progress being made. I think that as fellow committee members I've raised this on a number of occasions with a number of ministers and cabinet secretaries so we will be keeping on that I think so thank you for the answer Thank you very much Paul for having us online. Thank you just to extend some of the conversation that we've just had and colleagues have already raised the point of fair work and of course it's welcome that the fair work criteria has been extended to all staff engaged in grant funding activity but the sector's really really struggling and I know or I believe the cabinet secretary knows that and understands that all departments or local government to resource voluntary organisations through grants to pay at least the real living wage Pam Duncan-Glancy makes a fair point and every part of the budget every whether it's local government, whether it's third sector, whether it's public agencies these are tough times and it's going to be tough for all sectors in terms of the quantum of finance available so the importance of multi-year funding is a question of what can we do to make things easier to give that certainty of funding availability and I think moving to a position where multi-year funding is the presumption and the default will help of budgets around that in terms of going forward we will work with the third sector to make sure that we can support them as much as we can we need to look at how we can reform the way we do things that doesn't just apply to Scottish Government agencies and public services to local government and I know that the Account Commission had suggested that as a way forward and may have to apply to the third sector I think that we all have to look at for example the buildings that we use the services and support structures that we use can they be shared for example in ways that perhaps we need to look at things a bit differently I think that the pandemic has shown that there are ways of working that are different that perhaps have been achieved in a very short time frame that perhaps pre to pandemic we wouldn't have thought possible in terms of flexible working which in some ways can reduce some of the infrastructure costs that organisations have so these are the types of discussions that we need to continue to have not going to sit here and pretend that it is easy for any sector including the third sector and as I say we will try to support them as much as we can I thank Cabinet Secretary for that answer and I find it difficult to not come to the conclusion that that effectively means that a policy that the Scottish Government can say that we would support as well in fair work is now going to mean that third sector organisations are going to have to make cuts so that really does feel quite disappointing I understand the context that we're in but those organisations as the Cabinet Secretary rightly pointed out created not just flexibility during the pandemic but also where the organisations who stepped up on the front line when others couldn't so I really would urge the Cabinet Secretary to make a commitment to resource the demands that she and her Government make of third sector organisations so that they're able to meet them and just on the point of long term funding I wonder if the Cabinet Secretary would be able to set out what timescale she's currently working to for delivering the multi-year funding and would she determine that multi-year funding as a three-year minimum commitment First, just on your previous point I mean there is I think an argument and the third sector has made this quite often that being able to pay your paying staff the real living wage and being able to hold on to staff is probably quite more cost effective than having a high turnover of staff and that's not to underestimate the challenge but I think having that multi-year funding of certainty to be able to know that staff can be that they're going to have a contract next year prevents people leaving to look for other jobs because their funding is uncertain so that's why I think the importance of multi-year funding and it's probably been the key ask from the third sector above all else you can't be underestimated I think where we expect to be working with a stakeholder on a long-term basis I expect the grant they receive to be on a multi-year basis so as I said earlier we want this to become the kind of default it's of course not always appropriate where you might have a project that's coming to an end or it might be a very short-term funding arrangement that they've been given like a one-year one-off proposals now and we will subsequently monitor progress across the fairer funding principles including multi-year funding to make sure that we can roll that out as quickly as we can but it will become the default and the presumption rather than the exception In fact, Cabinet Secretary for that answer, do you know when it will become the default rather than the exception? Well, as I say, the officials are processing grant app proposals now and we would expect that unless those are short-term funding or projects coming to an end that multi-year funding will be getting put in place from now on for as many organisations as possible now there may be some reasons and some that that can't be done but I would assume that from now on that will be the default but happy to keep the committee appraised of progress in terms of this funding round when those decisions are made if that would be helpful we can come back with the details Thank you, Cabinet Secretary I just have one further question and I'll be brief Yesterday I met with Patec Thistle Trust accepting activity group and they said that the financial support they got had underestimated the energy costs by hundreds of thousands of pounds and we know from evidence to this committee that a number of third sector organisations are feeling just as much financial difficulty as the result of the costs of living as the people they are supporting funding we don't believe and they don't believe has taken account of that and security so can the Cabinet Secretary set out whether she intends to introduce any cost living measures that would help organisations to meet those costs and what could I tell the Patec Thistle Trust and others around whether or not there is an intention to increase the offer to those organisations to account Well, first of all if Pam Duncan-Glancy wants to write to me with the details of the organisation that she's referring to I'm happy to come back with a more detailed response but in principle we have tried to support third sector organisations with some of the costs of living pressures but I think I would have to be honest and say I don't think whether it's for organisations or households that we are able to mitigate every single increase of cost including energy costs we just don't have the resource to be able to do that in every circumstance we have tried to help where we can but we just can't mitigate every single increase in cost for every organisation we just don't have the resource to do that but if Pam Duncan-Glancy wants to write to me with the specific details of that case I'm happy to look into it in more detail Pam Duncan-Glancy it's very positive news on the multi-year funding this is something that's been raised with us frequently in the committee so it would be really helpful to keep us updated on that okay we'll move now to questions from Jeremy Balfour just of a final point cabinet secretary, you appreciate I've asked you and asked the cabinet secretary to finance on a number of occasions in regard to specific discussions with SCVO around three-year funding and I know from previous answer you and your officials do have regular meetings with them I'm just wanting to clarify had this been discussed specifically with SCVO three-year funding and what on-going discussions are planned on this specific issue with them officials meet with other umbrella organisations if you like for third sector organisations on a regular basis I can't remember exactly when I met with the chief executive of SCVO but I do recall having a discussion last year third-set multi-year funding is always raised in these discussions understandably so as a key ask and we'll continue to work with SCVO on the moving to this new system of multi-year funding our relationship with SCVO and others is good they've been an extremely important partner at the table around Covid recovery so other ministers clearly have met with them as well around the focus of the input there and we'll continue to to do that, it's an important relationship thank you I should just mention there of other organisations aren't funded just on your budget, we're funded through budgets across different Government portfolios so I wonder if you could just outline what efforts are made across the whole of Government to adopt a fair funding practice outlined by SCVO and for my information who coordinates that if that's coordinated by the finance team or that's coordinated within your administration to make sure that organisations don't miss out because one department doesn't give them money and another one does or the other way reverse so we would expect officials across Scottish Government to be working closely on this so knowing what grant applications are in which ones are likely to be approved or not approved and sometimes it's important if there is a particular fund that is under real pressure whether there's other funds within Government that can be brought to the table to support and that happens on a fairly regular basis so it shouldn't matter where the fund is located within Government, the same approach should be taken in terms of that move to multi-year funding and also the key priorities that have been set out so the Deputy First Minister set out those key priorities of tackling trial poverty, of sustaining public services of moving to net zero you would expect those priorities to be seen throughout all of the funds for the third sector across the various parts of Government is it right in every single circumstance? I'm sure there will be glitches on occasion but that is how we would expect it to operate and we would expect that close communication to take place Thank you My final question for today is around the money that's given to local authorities to pass on to local third sector organisations in their local area ultimately up to them but do you recognise the budget that local authorities now have to give on to third sectors has in real terms gone down and from what we're seeing across Scotland a number of third sector charities are going to really struggle to get any funding from local authorities this year, I may have to close because of a real cut in the budget for local authorities I could probably spend a good half hour talking about local government finance and we had a long discussion at the committee on Tuesday around this but the £570 million that has been allocated to local government I think in the context of only £800 million the entire of Barnett consequentials coming is tough and it's difficult and challenging but perhaps in that context not an unreasonable settlement and the accounts commission verified that figure of £570 million the accounts commission also pointed to a couple of other things they pointed to reserves and said that local authorities should keep under constant review the priorities for the use of those reserves, some of the decisions around the priorities of individual local authorities might have been taken some time ago and in the midst of a cost of living crisis those should always be kept under review the other thing that the accounts commission pointed to was reform and I mentioned earlier on the need for us to all look across the public sector particularly when you look towards years three and four of this budgetary cycle of what is coming down the line from the UK Government in terms of the projected finance this is the time to be making moves towards reform because years three and four are very very challenging indeed now we are trying to give local authorities maximum flexibility so that they can make the decisions about where they spend their money by removing as much ring ffencing as possible and that would then be down to each local authority to decide where it spends its money and of course trying to support revenue raising opportunities and of course they have complete freedom around council tax levels I guess we come back low to the previous exchange I had with Miles Briggs around some discrete areas of funding that we would want to see local authorities continue to spend in terms of those joint priorities so these are not easy things to balance but we would hope and expect that local authorities would see the value and value for money in the third sector investment that they make in terms of the many projects that are delivered are very good value for money and I think local government recognises that as much as we do we will continue to have these discussions with both local government and the third sector but the context in the backdrop to this has to be acknowledged I think We will move now back to Emma who will finish our questions of for today We have taken an interest as a committee in how the government overall uses human rights budgeting and the secretary gave an example of where spending allocations have been changed following inequalities in human rights assessment and advised how we as a committee but also the wider public can see how human rights considerations have affected budget decisions It is an important question there are a number of on-going processes happening across portfolios as each continues to assess the value and impact of policies the vast majority of the programmes, policies and services that the budget funds are not necessarily freshly generated and announced at the point at which the annual budget is introduced and budgets are more likely to be reviewed and refined in the light of a number of factors including the human rights dimension a number of the large policy areas have substantive evaluation programmes in place to help evaluate the outcome of their spending programmes and these deliver results at different points in the policy cycle and we would expect the equality and human rights assessments to be a key part of that they are not the only factors to inform the final decisions but they are an important part Other considerations may include making budget adjustments due to changes in available resources to support the policy delivery reprofiling of the phasing of delivery a reprioritisation and as we did in the emergency budget review of focusing down on other key priorities and that means perhaps not funding other things I think that the equality and human rights assessment is key whether of itself is a reason for changing a budget decision is probably not but it would be part of the assessment of the priority of that funding decision That's really helpful and looking at the equality and fair Scotland budget statement the right to an adequate standard of living is explicitly listed as a key focus in almost every portfolio not just the cabinet secretary is that a reflection of Scottish Government priorities as well as these overall efforts to realise human rights Well it is and I would hope that that is seen through whether it's the £3 billion allocated for helping low income households a billion of which is only available in Scotland over this last year and going forward the budget decisions that we have made not least that £780 million above the block grant adjustment for social security is an active political choice and decision made around how do we support household budgets and those on low incomes now that comes at a cost in terms of the availability of funding for other things but in the current financial climate I think it is absolutely the right decision to make and I think one that we would defend vigorously Thank you very much Emma. Alright thank you cabinet secretary and thank you to your officials for joining us this morning we will have a short five minute comfort break just to allow the cabinet secretary to leave Hi welcome back everyone so we will move on to agenda item 3 which relates to the committee's child poverty and parental employment inquiry to assist the committee with consideration of its next steps we are invited to consider and agree to discuss further approaches, evidence received, correspondence and draft reports in private at future meetings are we all agreed we are all agreed we will now move into private session can members who are joining us remotely please use the teams link in their calendars to join the meeting thank you


Welcome back. We are trying to prove the Cauchy-Piano existence theorem, where we use only the continuity assumption on the function f. To prove the piano theorem, first we prove the existence of an approximate solution to the initial value problem. Let us recall the theorem we have stated, that is theorem 1. Consider an initial value problem d over dx with initial condition d over dx is f of x y with initial condition y at x 0 is y 0, where the function f is continuous on a rectangle defined by x minus x 0 is less than equal to a and y minus y 0 is less than equal to b. And we take some constants m is the maximum value of f in the rectangle and h is minimum of a and b by m. Then the conclusion of the theorem is that there exists for a given epsilon greater than 0, there exists an epsilon approximate solution to the initial value problem on the interval x minus x 0 less than equal to h. So, we will prove this theorem now. So, the proof is that for a given epsilon. So, let epsilon greater than 0 be given and let us consider the rectangle r. So, this is a rectangle r we defined, where this side is y minus b y 0 minus b and this side is y 0 plus b. This side of the rectangle is x 0 plus a and this side of the rectangle is x 0 minus a. And we are looking for a solution approximate solution on the interval x 0 to x 0 plus h. The given initial point is here x 0 y 0 is a given initial point. And now, we make polygons and join them to get an approximate shape of the solution curve. And approximate solution will be constructed and approximate solution. So, by approximate means epsilon approximate epsilon is given. Solution will be constructed for the interval x 0 x 0 plus h the right hand side to the given point x 0 and a similar construction will define it for the left hand side left interval that is for x 0 minus h to x 0. We will prove the epsilon approximate solution to this interval x 0 x 0 plus h. The approximate solution will consist of the epsilon approximate solution the solution will consist of. So, polygonal path starting from x 0 y 0 starting from the given point x 0 y 0 the initial point. So, it start from here this point x 0 y 0 there is a finite number of a finite number of straight line segments joined end to end to form the polygonal path that is a graph of the approximate solution.  So, therefore, this implies that for every epsilon. So, we start up with an epsilon for that epsilon greater than 0 there exist a delta of course, the delta will be a function of epsilon such that f of x y minus f of say x tilde y tilde and this difference is less than epsilon whenever x minus x tilde is less than delta and y minus y tilde is less than delta. So, this follows from the continuity in particular uniform continuity f on r. Now, what we do is now we divide the interval divide the interval on which we are looking for a solution x 0 x 0 plus h this interval we divide into n parts into n parts say x 0 is a first point then write it x 1 less than x 2 less than x 3 etcetera x n that is the last point which is x 0 plus h and make the maximum of x k minus x k minus 1 for any k which is less than or equal to minimum of delta which is delta we define just above delta epsilon and delta epsilon y m. So, the procedure is as follows. So, from x 0 y 0 the given initial point construct a straight line a straight line with slope f of x 0 y 0 which the slope we get from the differential equation we are starting from the point x 0 y 0 and we know the slope of the solution at the point x 0 y 0 is f of x 0 y 0 from the differential equation as d y by d x is f of x y. So, you construct a straight line with a slope f of x 0 y 0 proceeding to the right of x 0 to the right of x 0 until it intersect the line x is equal to x 1. So, what we do is we start from the point x 0 y 0 and with slope f of x 0 y 0 draw a straight line let it touch the line x is equal to x 1 at this point. So, we got a line segment having slope f of x 0 y 0. So, let the line this line intersect the line x is equal to x 1 at some point say at the point x 1 y 1 x 1 is the point right to x 0 and y 1 is the point at which the straight line hits the line x is equal to x 1. Now, this line segment the line segment lies inside the triangular portion inside the triangular region bounded by the lines by the lines starting from x 0 y 0 with the slopes m is equal to a slope m and minus 1. See this line segment which we just drawn from x 0 towards the right of x 0 that is going up to x 1 and this line segment is inside the triangular portion this is a triangular portion triangular region.  So, it is a triangle formed by three lines and this approximate solution portion of the approximate solution starting from x 0 y 0 lies inside this triangular portion and why what is the guarantee that this line segment will lie inside the triangular portion. So, this follows. So, from the definition of h which is minimum of a and b by m and the fact the bound for f of x is h plus m f is bounded by m. Now, we define solutions on each of these sub intervals. Now, we define solutions we now define approximate solution each sub interval in the same way. So, in the first interval that is in the interval x 0 x 1 is the first interval there the solution is y of x is equal to starting from y 0 plus slope slope is f of x 0 y 0 and this is up to the and x varies up to x 1 this is from x minus x 0 this is a straight line equation of a straight line having slope f of x 0 y 0 and starting from the point x 0 y 0 this solution on the interval x 0 x 1 and similarly, so on the interval x 1 x 2 the solution is y of x is equal to and there we start from the last point of the previous interval which is y at x 1 which we obtained from the first a solution of the first interval y at x 1 plus the next slope is f of x 1 and the point there is y of x 1. So, this is a slope and x minus x 1. So, this is a solution defined on it is a straight line starting from the last point x 1 x 1 y 1 that is x 1 y at x 1 and similarly, if we keep on doing on the interval x k minus 1 x k on this interval. So, y x is given by the solution given by y of x k minus 1 plus the slope at y k minus 1 f of y x k minus 1 into x minus x k x k minus 1. So, this in general is a solution on any kth interval of 2 points x k minus 1 x k where k is equal to 1 2 3 etcetera. So, if we join all these solutions these are the solutions straight lines we obtained on each of the interval and if we join them and we get a function which is having a piecewise derivative piecewise continuous derivative. So, the function y obtained on the interval x 0 x 0 plus h is a function having piecewise continuous derivative. So, that is y is an element of c 1 p x 0 x 0 plus h. So, in the property of approximate solution we want the solution to be in c 1 p on those points at which the derivative are defined. Let us consider, so for any x and x tilde let us take 2 points x and x tilde in x 0 x 0 plus h. To study further properties of y let us take 2 points x tilde and x tilde x and x tilde in the interval x 0 and x 0 plus h. Without loss of generality let us assume that x c x tilde is an element in one of the sub intervals. So, it is in x i minus 1 x i and x is in x j minus 1 x j and assume that this is x i minus 1 this is x i. So, this is x j minus 1 this is x j and we take x from here. So, x tilde is somewhere in the interval between x i minus 1 and x i and x is from the interval x j minus 1 x j. So, consider the difference y of x minus y of x tilde y of x minus y of x tilde. This we can write this as y of x minus y of x j minus 1 write it to fresh y of x minus y of x tilde is equal to y of x minus y of x j minus 1 plus y of x j minus 1 adding and subtracting j minus 1 minus y of i plus y of x i minus let us consider y of x minus y of x tilde is equal to y of x minus y of x j minus 1 plus y of x j minus 1 minus y of x i plus y of x i minus y of x tilde if you group them. So, this is less than or equal to y of x minus y of j minus 1. So, remember y x j minus 1 is less than x plus x tilde plus y of x i minus y of x tilde and if you look at the properties of the solution the function y because y is a solution and y is a solution satisfied by the formula given that is the equation of a straight line. So, this is equal to less than or equal to m times x minus x j minus 1 plus x tilde. So, remember the form for y x is equal to y of x k minus 1 or in this case j minus 1 plus f of x j minus 1 y x j minus 1 y of x is y of x j minus 1 plus f of slope x j minus 1 y x j minus 1 into x minus x j minus 1 and we know that by definition this quantity. So, there is if you take the difference y x minus y x j minus 1 is less than or equal to the slope x j minus 1 y x j minus 1 into x minus x j minus 1 and this quantity is bounded by this is bounded by m. So, therefore, we get the first time is less than or equal to m times x minus x j minus 1 and similarly the other times are coming as a solution from the each interval we can compute it to get this is also m times x i minus x tilde. Now, use of fact that this x 1, x 2, x 3 they are all in the increasing order. So, therefore, this absolute value we can get rid of that this is less than or equal to m times x minus x j minus 1 plus x j minus 1 minus etcetera minus plus x j x i minus x tilde and we will be able to cancel out these terms to obtain m times absolute value of x minus x tilde. So, what we have shown is the absolute value of y of x minus y of x tilde where x and x tilde are from anywhere in the interval x 0, x 0 plus h is less than or equal to a constant m times x minus x tilde. So, in particular now, so you call this inequality as star we will be using it again. So, in particular let x tilde is equal to x 0 the first initial point x 0 then what we have is y of x minus y of x 0 which is less than or equal to m times x minus x 0. So, that is absolute value of y of x minus y of x 0 is y 0 is less than or equal to m times x minus x 0 is bounded by h and which is again bounded by b by definition of m by definition of h this is bounded by m. So, therefore, what is the conclusion the conclusion is. So, we obtained y x for any x minus y 0 is less than or equal to b. So, therefore, this implies that x y of x this is a point inside r for x in the interval x 0, x 0 plus h. So, hence f of x y of x is defined y of x is a approximate solution we defined we constructed and f of x y x is defined for all x for x in the interval x 0, x 0 plus h. Now, we consider an x in the interval say x k minus 1 x k a particular interval. So, we get also we know that in this interval the solution y x is given y.  This is a equation of the straight line or the solution defined in the interval x k minus 1 x k. Obviously, f prime x is a slope is f of x k minus 1 y of x k minus 1 which we will use later. So, let us see what is y of x minus y of x k minus 1 here y of x the solution minus y of x k minus 1 the solution at the starting point. So, this is less than or equal to if you make use of this equation y of x minus y of x k minus 1 is bounded by the bound of f which is bounded by m. So, this is m times x minus x k minus 1 which is of course, less than by definition of our partition m into delta y m which is delta. So, that delta is a function of epsilon. Now, if we take the difference of the derivative y prime x minus f of x y x. So, in this interval if you find the difference of this solution the derivative of the solution with the right hand side of the differential equation y prime x minus f of x y x. In fact, this is going to be the difference between the solution. So, which is equal to we have seen that y prime x is f of x k f of x k y x k minus 1. So, this is f of x k minus 1 y x k minus 1 minus f of x y of x and from the uniform continuity of f we have seen that. So, this is less than or equal to epsilon by the uniform of f. So, whenever x minus x k k minus 1 and x the difference between x k minus 1 x is less than delta and y x k minus 1 and y x less than delta this difference is less than epsilon by the uniform continuity of f. So, what does it say? So, this implies that if y now satisfies all the properties of an epsilon approximate solution. So, this implies that y x. So, y x is an epsilon approximate solution. So, y is in c 1 p and y prime minus y this is less than epsilon for all points at which the derivatives are defined. So, that is not satisfied only at a finite number of points at which the derivatives having gem type discontinuity. Now, with these background we are going to prove the celebrated theorem of Cauchy and Piano. So, Cauchy Piano theorem on the existence of solution Cauchy Piano existence theorem. So, in the previous theorem under the same conditions of the previous theorem we are going to show that the initial value problem is going to have a solution. So, the statement of the theorem is let f x y be continuous on the rectangle r which is defined as set of whole x y such that x minus x 0 is less than or equal to a y minus y 0 is less than or equal to b. Then there exists a solution to the initial value problem d y by d x is equal to f of x y y at x 0 is y 0. In the interval x minus x 0 is less than or equal to h where h is minimum of a b by m with the m being m is equal to the maximum of the function f x y where x y is in the rectangles. So, for all x y rectangle if m is a bound is a maximum of it then there exists a solution in the interval starting from the point x 0 y 0. So, let us quickly see the proof of it by using the ideas from the previous proof, previous theorem proof. Now, with epsilon n to be 1 by n where n is equal to 1, 2, 3, etcetera these small sequence of positive numbers. Now, for each epsilon n for each epsilon n there exists epsilon approximate solution called it y n x to the initial value problem. So, this is true because of our previous theorem for every epsilon n there exists an epsilon n approximate solution which we denote by y n to the initial value problem. Now, by theorem the previous theorem 1, the theorem we have shown that y x and y n x the approximate solution y n x minus y 0 is less than or equal to b or y n x is bounded by y 0 plus b. So, right hand side is independent of n. So, therefore the family of solution. So, therefore this implies that y n x the solution this sequence of solution is uniformly bounded. So, this is a uniformly bounded sequence of functions again from the previous theorem from the previous theorem. In fact, from the inequality from star if you look into the previous theorem we have shown that this is star this inequality. So, from star that y minus y x minus y x tilde is less than or equal to m x minus x tilde for all x and x tilde in the interval x 0, x 0 plus h. So, therefore again from star we have that y n x minus y n x tilde is less than or equal to m times x minus x tilde where x and x tilde are numbers from x 0, x 0 plus h. So, what does it say? It says that. So, this implies that the sequence y n is an equicontinuous. So, equicontinuous functions. So, therefore the approximate solutions epsilon n approximate solutions which are denoted by y n's. So, y n's are uniformly bounded and equicontinuous. Now, we invoke Arcella-Oscoli theorem. We here make use of Arcella-Oscoli theorem. So, thus by Arcella-Oscoli theorem there exist a subsequence of y n, a subsequence y n say y n k. So, there exist a subsequence of y n k such that y n k converges to some function y uniformly x 0, x 0 plus h. So, invoking Arcella-Oscoli theorem we get a sequence, a subsequence of the constructed sequence y n that converges uniformly to some function y in the interval x 0, x 0 plus h. So, this implies that y is continuous. So, again by using the theorem we discussed in the preliminaries a sequence of function converges uniformly and each y n k is continuous, then the limit function is also continuous. And y of x minus y of x tilde is less than or equal to m of m times x minus x tilde. Now, our task is to prove that this limit function y is a solution to the initial value problem. So, we now prove that the limit function y satisfies the initial value problem. So, for that consider the error term. So, consider the term denoted by delta n k of x which is equal to y n k which is a subsequence we extracted from the sequence prime minus f of x y n k x. So, this difference is this error if y prime n k exist, otherwise we take this 0, otherwise that is otherwise case is only for finitely many the many points. So, in other words this is a y n k y n k prime is equal to f of x y n k x plus delta n k x. Now, integrating over x 0 x we get. So, integrating we get y n k x k x is equal to y 0 plus integral x 0 to x f of t y n k t plus delta n k t d t. And we know that y n k converges to y uniformly. So, this implies that as f is continuous f of t y n k t converges f of t y t uniformly. So, uniformly as f is continuous f is uniformly continuous. So, this implies that the integral x 0 to x f of t y n k t d t converges to integral x 0 to x f of t y t d t. Further we have delta n k which is by definition is less than epsilon n which is 1 by n and passing to the limit this tends to 0. So, we prove that we prove that the limit of star is y of x is equal to y 0 plus integral x 0 to x f of t y of t d t. And obviously, y at x 0 is y 0 and by basic lemma y has to satisfy. So, by basic lemma y satisfies the initial value problem. So, therefore, the limit of this convergence of sequence y n k is a solution to the initial value problem. So, this proves that solution exist thus the limit function y n is a solution to the initial value problem. So, in this lecture we have proved that if we have an initial value problem d over d x is equal f of x y and f is continuous with respect to x and y then there exist a solution to the initial value problem. The uniqueness is not guaranteed.