Convergence of a Recursive Sequence

[life is maths]

Homework Statement

A sequence {a_n} defined recursively by a₁=1 and a_n+1=\frac{1}{2+a subn}, n\geq1. Show that the sequence is convergent.

Homework Equations

If a sequence is bdd below and decreasing or it is bdd above and increasing, then it is convergent.

The Attempt at a Solution

{a_n}\geq0, hence it is bounded below. I checked a few terms from the beginning and obtained a decreasing sequence. I tried induction to show this, but it didn't work. Base case is O.K. a₀>a₁ Suppose a_k>a_k+1.
Then we get
a_k+2=\frac{1}{2+a sub(k+1)}>\frac{1}{2+a subk}=a_k+1, and a_k+2>a_k+1. I feel like I'm doing a stupid mistake, and I can't understand, if this is true, why the induction does not work. I would be grateful if you could help me. Thanks for your time and effort :)

 

[micromass]

Induction doesn't always work since the sequence is not descreasing. If you check more terms then you see that.

Are you familiar with the contraction mapping theorem by Banach?

 

[life is maths]

Induction doesn't always work since the sequence is not descreasing. If you check more terms then you see that.

Are you familiar with the contraction mapping theorem by Banach?

Ah, really? So that was my mistake... I have no idea about contraction mapping theorem.
Thanks for enlightening me
Is there a more practical way to show a sequence is convergent?

 

[life is maths]

Can I use the limit definition? I see no way out... :(

 

[gopher_p]

You may be able to show that the "odd" elements of the sequence form a decreasing subsequence and the "even" elements an increasing subsequence ...