Recursion relation convergence

[hasan_researc]

Homework Statement

Consider the sequence $$\left x_{n}\{\right\}$$ defined by the recursion relation,

$$x_{n+1} = \frac{1}{2} \left( x_{n} + \frac{2}{x_{n}} \right)$$

where x₀ > 0.

Use the fact that "if a sequence of real numbers is monotonically decreasing and
bounded from below, then it converges" to prove that the sequence converges.

Show that it converges to $$\sqrt{2}$$.

Homework Equations

The Attempt at a Solution

No idea!
Any help would be greatly appreciated.

 

[HallsofIvy]

No ideas? How about the idea of doing what the hint says?

1) Prove the sequence is increasing. Use "proof by induction" to show that $x_{n+1}\ge x_n$ for all n.

2) Prove that the sequence has an upper bound. Again, use "proof by induction to show that $x_n< 2$ for all n.

 

[hasan_researc]

This is what I have found so far:

$$x_{n+1} < x_{n}$$
$$\frac{1}{2}\left( x_{n} + \frac{2}{x_{n}} \right) < 2x_{n}$$
$$x_{n} > \sqrt{2}$$

But I have n't used induction to show that the sequence is bounded below by $$\sqrt{2}$$

Now, I have to show that the sequence is monotonically decreasing (by induction, as you say).

But how I prove the $$x_{n+1} \leq x_{n}$$ for n = 1 (to begin with)?