How to prove that a revursive sequence is convergent or not, and if it is, how to find its limit? Suppose $a_1 = 2$, and
  $$ a_{n+1} = \frac{1}{2}\left(a_n + \frac{2}{a_n}\right) $$
  if $n \ge 1$.
  Is the sequence convergent or divergent?
  If it's convergent, find
  $$\lim_{n\rightarrow\infty}{a_n}$$


Usually, what's the approach we use for recursive sequence like this?
A friend of mine told me that any recursive sequence is divergent, but after I checked on calculator, it seems that the series is approaching one particular value.
First of all, Welcome to the site !

If the limit exists, it is given by
$$L = \frac{1}{2}\left(L + \frac{2}{L}\right)\implies L=\pm \sqrt 2$$

Now, your turn !
First of all, Welcome to the site !

If the limit exists, it is given by
$$L = \frac{1}{2}\left(L + \frac{2}{L}\right)\implies L=\pm \sqrt 2$$

Now, your turn !