Do All Bounded Monotone Sequences Converge?

[kmikias]

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.

 

[kmikias]

Additional to the question i asked,

a_{N(E)} < L - E
THEN
a_{(N(E)} < L - E < a_{n}
but we know
a_{n} >= L
THEN
L <= a_{n} <= L+E

BUT I STILL HAVE PROBLEM BECAUSE I DON'T KNOW IF a_{n} <= L+E IS TRUE.