# Sequences and existence of limit 2

1. Nov 12, 2012

### Felafel

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

Let an be a bounded sequence and bn such that

the limit bn as n→∞ is b and

0<bn ≤ 1/2 (bn-1)

Prove that if:

an+1 ≥ an - bn,

then

lim an
n→∞

exists.

2. Relevant equations

3. The attempt at a solution

as 0<bn ≤ 1/2 (bn-1) the sequence bn is decreasing.
thus its limit, b, is 0 (can i assume that?)

Then, assuming an converges to a limit, say p, the equation is:

p ≥ p - b where b=0

p ≥ p is true for every p.

Then, an is constant and therefore converges.

Does is it work like that?

2. Nov 12, 2012

### SammyS

Staff Emeritus
You can't assume that an converges. That's what you need to prove.