real analysis - Limit $(n - a_n)$ of sequence $a_{n+1} = sqrt{n^2 - a_n}$

Consider the sequence $\{a_n\}_{n=1}^{\infty}$ defined recursively by
$$a_{n+1} = \sqrt{n^2 - a_n}$$ with $a_1 = 1$. Compute $$\lim_{n\to\infty} (n-a_n)$$

I am having trouble with this. I am not even sure how to show the limit exists. I think if we know the limit exists, it is just algebra, but I'm not sure.