I have to prove the following sequence is bounded (Or prove otherwise)
$$
a_{n+1} = a_n + \frac{1}{a_n}
$$
When $a_1$ is any real positive number.
I have tried using induction. But I'm not sure if what I'm getting to is correct.
My take:
Assume $a_1$>1 (if not, then in the next step $a_2$ will be – and the proof is the same).
First we show for base case:
$$
a_2 = a_1 + \frac{1}{a_1} < 2a_1
$$
Assume by induction that this holds for $a_n$:
$$
a_n < 2a_1
$$
Here is where I got stuck. Any algebra manipulation I tried to do left me with a leftover of which I can't seem to get rid of, or an inequality which doesn't do any considerable progress.
For instance:
from the above we get:
$$
\frac{1}{a_n} > \frac{1}{2a_1}
$$
$$
a_n + \frac{1}{a_n} > a_n + \frac{1}{2a_1}
$$
Which doesn't help.
Perhaps this sequence isn't bounded?
I'd appreciate any help at all (preferably a nudge) 🙂
Best Answer
First, prove inductively that $a_n > 0$ for every $n$. You deduce that $$a_{n+1}-a_n = \frac{1}{a_n} > 0$$
so the sequence is increasing.
So either it is unbounded, either it converges. If it converged to a limit $l \in \mathbb{R}$, you would get $$l=l+\frac{1}{l}$$
so $$\frac{1}{l}=0$$
which is impossible. So the sequence is unbounded.