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## Homework Statement

A bounded monotone sequence converges.

Proof

for bounded monotone increasing sequence and decreasing sequence.

Does both them converges?

## Homework Equations

So, I used the least upper bound and great lower bound to prove increasing sequence and decreasing sequence.

Property of LUB and GREAT LOWER BOUND.

## The Attempt at a Solution

a bounded monotone increasing sequence to converge....

Proof.

a_{n} is monotone increaing if n > N(ε), then a_{n}≥ a_{N(ε)} > L -ε. But a_{n) ≤ L.

thus L - ε < a_{n} ≤ L for n > N(ε); that is | a_{n} - L | < ε for n>N(ε). Δ

Proof for a bounded monotone decreasing sequence to converge..

this is where i got lost.

so i used great lower bound to do the proof.

we know G.L.B has this two property

1. a_{n} ≥ L for every n

2. for ε > 0, there exist a positive number N(ε) SUCH THAT a_{N(ε)} < L-ε

so

a_{n} is monotone decresing if n > N(ε), then L ≤ a_{n} ≤ L +ε. am kind of lost here.

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