Recursion relation convergence

[hasan_researc]

Homework Statement

Consider the sequence [tex]\left x_{n}\{\right\}[/tex] defined by the recursion relation,

[tex] x_{n+1} = \frac{1}{2} \left( x_{n} + \frac{2}{x_{n}} \right) [/tex]

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 [tex]\sqrt{2}[/tex].

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 [itex]x_{n+1}\ge x_n[/itex] for all n.

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

 

[hasan_researc]

This is what I have found so far:

[tex]x_{n+1} < x_{n}[/tex]
[tex]\frac{1}{2}\left( x_{n} + \frac{2}{x_{n}} \right) < 2x_{n}[/tex]
[tex]x_{n} > \sqrt{2}[/tex]

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

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

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