Take the 2-minute tour ×  $a \in \mathbb R$ has the decimal
expansion $a = a_0.a_1a_2a_3 \ldots a_n \ldots$  Find all values
for $a$ for which the sequence $\{a_n\}_{n=1}^{\infty}$
converges.  I rule out irrationals first, because if they can't
be represented with integer numerator and denominator, they
can't have convergent decimal expansions. But how do I separate
the rationals, say differ between say $1/7$ and $1/3$?
share|improve this question add comment  3 Answers  up vote 2
down vote accepted  As you have observed since $a_k$'s are
digits from $0$ to $9$, it will converge only when there exists
an $N$ such that $a_n = d \in \{0,1,2,3,4,5,6,7,8,9\}$ for all
$n>N$.  (To see this, choose say $\epsilon = 0.1$. Since $a_n$
converges, this means there exists $N(\epsilon)$, such that
$\vert a_{n+1} - a_n \vert < \epsilon$ for all $n > N$. But
$a_n, a_{n+1} \in \{0,1,2,\ldots,9\}$. Hence, $a_{n+1} = a_n$.)
Hence, $$a = a_0.a_1a_2\ldots a_N dddddd \ldots$$ $$10^Na = a_0
a_1 \ldots a_N . dddd\ldots$$ $$10^{N+1