TITLE: Showing a recursive sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$
QUESTION [6 upvotes]: Show the sequence isn't bounded: $a_1=1$, $a_{n+1}=a_n+\frac 1 {a_n}$.

Proof by contradiction: Let $M>0$ such that $\forall n: |a_n|< M$. 
Let $\epsilon >0 $ and for some $n=N, \epsilon: a_N=M-\epsilon<M $ pluging that in the recursion: $a_{N+1}=M-\epsilon+\frac 1 {M-\epsilon}>M>M-\epsilon$. Contradiction.
I wondered if I could suppose about the boundary that $\forall n: |a_n|\le M$ ? The proof would basically be the same only I could drop the epsilon.

REPLY [5 votes]: Somewhat of a funny proof.
Suppose for contradiction that $a_n$ is bounded by some positive $M$
Then the series $\displaystyle \sum (a_{n+1}-a_n)=\sum\frac{1}{a_n}$ has bounded partial sums (and positive general term).
Thus, the series $\displaystyle \sum\frac{1}{a_n}$ converges, which implies $a_n\to \infty$
Contradiction.