Why is $a_{1} > 0 \land a_{n+1}=a_{n}+\frac{1}{a_{n}}$ unbounded? Let $(a_n)$ be a sequence s.t $$a_{1} > 0 \land a_{n+1}=a_{n}+\frac{1}{a_{n}}$$
Prove that $a_{n}$ is unbounded.
Proof:
Consider $a_{n+1}−a_{n}$:
$a_{n+1} - a_{n} = a_{n} + \frac{1}{a_{n}} - a_{n} = \frac{1}{a_{n}}$.
This is greater than $0$. Thus, $a_{n}$ is increasing.
It was proved that $a_{n}$ is increasing. Assume that it is bounded. Then it would follow that $a_{n}$ is convergent to a real number $L>0$. But taking $n\to\infty$ into the recurrence relation gives
$$L+\frac{1}{L} =L$$
which is a contradiction. Therefore $a_{n}$ is unbounded
I found this on the site but I don't get why it is unbounded. Could someone plz explain?

Note that $$a_{n+1} - a_n = a_n +\frac{1}{a_n} - a_n = \frac{1}{a_n}>0$$ since $(a_n) > 0 $ for all $n$. Therefore the sequence is strictly increasing. 
So we either have: 


*

*the sequence converges and is bounded (can you prove this?)

*the sequence does not converge and is unbounded (can you prove this?)


Assume it does converge. Then, we know that the limit $l$ must be greater than 1 or equal to 1. By the shift rule, if the limit exists, it satisfies $$l= l+\frac{1}{l}$$ which is impossible. Then the sequence does not converge. 
Therefore, we have a strictly increasing sequence that does not converge. It must therefore be unbounded. 
