# Bounded sequence converges.

• kmikias
In summary, the conversation discusses the convergence of a bounded monotone sequence. The proof for a bounded monotone increasing sequence involves using the least upper bound and great lower bound, while the proof for a bounded monotone decreasing sequence involves using only the great lower bound. The great lower bound has two properties: for every n, a_{n} is greater than or equal to the limit L, and for any positive number ε, there exists a positive number N(ε) such that a_{N(ε)} is less than L-ε. The proof involves showing that a_{n} is between L and L+ε for n greater than N(ε).

## 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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