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Bounded sequence converges.

  1. Nov 9, 2011 #1
    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.
     
    Last edited: Nov 9, 2011
  2. jcsd
  3. Nov 9, 2011 #2
    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.
     
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