Take the tour ×  This is a homework problem.  I tried proving
this by means of induction. Verifying the case $n=1$ is easy,
the only solution to $\frac{1}{x_1} = 1 $ is $x_1 = 1$,
therefore there is only one, and thus a finite number of
solutions. I am having some difficulties with proving that the
case of $N+1$ follows from the $N$'th case (for some $N \in
\mathbb{Z}_{>0}$), assuming that the sum only has a finite
amount of solutions in $\mathbb{Z}_{>0}$ (our induction
hypothesis). Can you help me with this?  I am equally interested
in any other way to prove that the given sum has only a finite
number of solutions, whether it can be done by means of
induction or not.  Thanks in advance!  share|improve this
question Hint: If every $x_n$ is strictly bigger than $2012$,
then the sum can't be $1$. –  student Feb 15 '12 at 16:10 Hint
#2: Try to prove (by induction on $N$) a stronger result, that
for any rational number $\frac{p}{q}$ the equation
$\sum_{n=1}^{N} \frac{1}{x_n} = \frac{p}{q}$ has finitely many
solut