Proving convergence of a sequence with a given property

Question

Let $$a_{n + 1} = \begin{cases} a_{n} + 1/n & \text{ if } a_{n}^{2} \leq 2 \\ a_{n} - 1/n & \text{ if } a_{n}^{2} > 2 \end{cases}$$

Show that for every index $n$, $\left|a_{n} - \sqrt{2}\right| < 2/n$, and use this property to show that the sequence converges to $\sqrt{2}$.

So, I did the first part of this problem by proving by induction. The base case is obvious, and then for the induction step I just split it into two cases, one for each condition. But, now I'm having trouble using this sequence converges to $\sqrt{2}$ using the property that I established.

I'm pretty sure that I need to do an $\epsilon-N$ verification, since this is a real analysis book, in the convergence section. I'm really not sure how I would go about it, though.

Answer 1

Hint: $2/n\to 0$ when $n\to\infty$.

For $\epsilon>0$, choose $n_0\in\mathbb N$ such that $2/n_0<\epsilon$. Then we have, for $n>n_0$, $$|a_n-\sqrt{2}|\le 2/n < 2/n_0 < \epsilon$$