real analysis - Proving a sequence converges.

Let $a_{1} >0$ and let $a_{n+1} = \frac{1}{2} ( a_{n} + \frac{1}{a_{n}})$ for all $n \geq 1$. Show that $a_{n}$ converges and find it's limit.

The Attempt: I am going to use the Monotone Convergence Theorem to be able to show the sequence converges. First I am going to show the sequence is monotonically decreasing by induction. However, I am having a hard time figuring out if the sequence is monotonically decreasing.

Please give me hints to solve the problem. Please do not solve the problem completely.

Thank you very much.