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Sequences and existence of limit 2

  • Thread starter Felafel
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  • #1
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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.

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?
 

Answers and Replies

  • #2
SammyS
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Science Advisor
Homework Helper
Gold Member
11,227
953

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 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?
 

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