Finding integers and the limit of a sequence - from a 100 sequence and more (Russian book) Consider a sequence $a_{1}=1$ and for every $k>1$ integer $a_k=a_{k-1}+\dfrac{1}{a_{k-1}}$.
a) How many positive integers $n$ are there, satisfying that $a_n$ be an integer?
b) Find the limit (if there is) of the $a_k$ sequence!
Please help! It looks like that for a) the answer is 2. Thanks in advance!

Let's prove that
$$
1 \le a_n \le n
$$
Since $a_1 = 1$ and $a_{n+1} > a_n$ we clearly have $a_n \ge 1$. We can prove that $a_n \le n$ by induction. We have $a_1 = 1 \le 1$ and induction step is
$$
a_{n+1} = a_n + \frac{1}{a_n} \le n + \frac{1}{a_n} \le n + 1,
$$
since $a_n \ge 1$.
About question a). If $a_n$ is an integer then $1/a_n$ is an integer only if $a_n = 1$. That gives us case $a_2 = 2$. Also by definition $a_1 = 1$. There is no other integers in this sequence. Let $a_n = \frac{p}{q}$ where $p$ and $q$ are coprime and $q>1$. Then
$$
a_{n+1} = a_n + \frac{1}{a_n} = \frac{p^2 + q^2}{pq}.
$$
So we must have $p^2 + q^2 = m\cdot pq$ for some $m \in \mathbb{N}$. But that's not possible if $p$ and $q$ are coprime, because $q$ must divide $p^2$ and $q > 1$. So there are only two indices ($n=1,2$) for which $a_n$ is integer.
Now regarding question b). Since
$$
a_{n+1} = a_n + \frac{1}{a_n} \ge \frac{1}{a_n} + \frac{1}{a_{n-1}} + \ldots + \frac{1}{a_1} \ge \frac{1}{n} + \frac{1}{n-1} + \ldots + 1 = H_n
$$
where $H_n$ is a harmonic series, which clearly diverges.
