(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A bounded monotone sequence converges.

Proof

for bounded monotone increasing sequence and decreasing sequence.

Does both them converges?

2. Relevant 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.

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Bounded sequence converges.

**Physics Forums | Science Articles, Homework Help, Discussion**