Limit of recurrence sequence I have to find a limit (or prove it doesn't exist) for the following recurrence sequence.
$a_1 = 2; a_{n+1} = \frac{1}{2}(a_n + \frac{2}{a_n})$
Now I know, in order to find the limit, I first need to prove that the sequence is monotonic and bounded. I've made a partial table of values and concluded that the sequence is decreasing, thus to prove monotonicity, I've written down:
$ a_{n+1} < a_n  \rightarrow a_n > \sqrt{2} $
And that's all I could think of. I don't think the inequality above proves anything so I don't know how to continue. I tried to calculate limit of the sequence by using limits of elements as follows:
$ \lim a_{n+1} = \frac{1}{2}(\lim a_n + \lim \frac{2}{a_n}) = a\Rightarrow a = \sqrt{2}$
But without proving monotonicity and bounding, there's no proof the limit exists at all.
Thank you for any help in advance.

one prove easy that $$a_n>0$$ for all $n$ then we have by $AM-GM$
$$a_{n+1}=\frac{1}{2}\left(a_n+\frac{2}{a_n}\right)\geq \sqrt{a_n\cdot \frac{2}{a_n}}=\sqrt{2}$$
