Convergence of a sequence

1. May 3, 2012

life is maths

1. The problem statement, all variables and given/known data

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

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

3. The attempt at a solution
{an}$\geq$0, hence it is bounded below. I checked a few terms from the beginning and obtained a decreasing sequence. I tried induction to show this, but it didn't work. Base case is O.K. a0>a1 Suppose ak>ak+1.
Then we get
ak+2=$\frac{1}{2+a sub(k+1)}$>$\frac{1}{2+a subk}$=ak+1, and ak+2>ak+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 :)

2. May 3, 2012

micromass

Staff Emeritus
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?

3. May 3, 2012

life is maths

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?

4. May 3, 2012

life is maths

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

5. May 3, 2012

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 ...