# Sequences and existence of limit 2

## Homework Statement

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.

## 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?

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## Homework Statement

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.

## 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:
You can't assume that an converges. That's what you need to prove.
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?