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Math Help - 2 Proof of Sequences

  1. #1
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    Exclamation 2 Proof of Sequences

    1) Prove that lim (Sn) = 0 iff lim|(Sn)| = 0.
    Note: the || are absolute values.

    2) Let (Sn) be a convergent sequence, and suppose that lim Sn > a.
    Prove that htere exists a number N such that n > N implies Sn > a.

    These are to be proved using the definition the limit of a sequence:
    For each epsilon > 0 there exists a number N such that
    n > N implies |Sn - L| < epsilon

    Thank you so much I am very stuck right now.
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  2. #2
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    Quote Originally Posted by tbyou87 View Post
    1) Prove that lim (Sn) = 0 iff lim|(Sn)| = 0.
    Note: the || are absolute values.
    If lim s_n =0 that means,
    |s_n|<e for n>N

    Hence if you chose n>N we have,
    ||s_n|-0|=||s_n||=|s_n|<e
    Thus,
    lim |s_n| = 0
    The same the other way around.

    2) Let (Sn) be a convergent sequence, and suppose that lim Sn > a.
    Prove that htere exists a number N such that n > N implies Sn > a.
    .
    Let lim s_n = L

    By hypothesis L>a thus, L-a>0 thus (L-a)/2 >0

    That means,

    |s_n-L| < (L-a)/2 for n>N
    Open absolute values,
    -(L-a)/2< s_n - L < (L-a)/2
    Thus,
    L-(L-a)/2 < s_n < L + (L-a)/2
    Thus,
    (L+a)/2 < s_n < (2L-a)/2
    The fist inequality says,
    (L+a)/2 < s_n
    Thus,
    (a+a)/2 < (L+a)/2 < s_n for n>N
    Thus,
    a < s_n for all n>N
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