Sequences and existence of limit 2

[Felafel]

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.

Homework Equations

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?

 

[SammyS]

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.

Homework Equations

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 a_n 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?