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Math Help - Two proofs regarding limits

  1. #1
    Senior Member Pinkk's Avatar
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    Two proofs regarding limits

    I just want to make sure I have done these proofs correctly.

    1) Let (t_{n}) be a bounded sequence such that there exists M such that |t_{n}| \le M for all n, and let (s_{n}) be a sequence such that lim\, s_{n} = 0. Prove lim\, (s_{n}t_{n}) = 0.

    PROOF:

    Assume the given conditions above. This means that there exists N such that, if \epsilon > 0 and  n > N, then |s_{n}| < \epsilon. In particular, there exists N_{1} such that if \epsilon > 0 and  n > N_{1}, then |s_{n}| < \frac{\epsilon}{M}. Hence, |s_{n}t_{n}| = |s_{n}||t_{n}| < \frac{\epsilon}{M}|t_{n}| \le \frac{\epsilon}{M}M = \epsilon, given that \epsilon > 0 and n > N_{1}. And so, lim\,(s_{n}t_{n}) = 0. Q.E.D.

    2)Suppose that (s_{n}) and t_{n} are sequences such that |s_{n}| \le t_{n} for all n and lim\, t_{n} = 0. Prove lim\, s_{n} = 0.

    PROOF:

    Assume the given conditions above. This means there exists N such that, if \epsilon > 0 and n > N, then |t_{n}| < \epsilon. Hence, |s_{n}|\le t_{n} = |t_{n}| < \epsilon, and so there exists N such that, if \epsilon > 0 and  n > N, then |s_{n}| < \epsilon. And so, lim\, s_{n} = 0. Q.E.D.

    Any feedback would be appreciated, thanks.
    Last edited by Pinkk; February 3rd 2010 at 07:26 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    I just want to make sure I have done these proofs correctly.

    1) Let (t_{n}) be a bounded sequence such that there exists M such that |t_{n}| \le M for all n, and let (s_{n}) be a sequence such that lim\, s_{n} = 0. Prove lim\, (s_{n}t_{n}) = 0.

    PROOF:

    Assume the given conditions above. This means that there exists N such that, if \epsilon > 0 and  n > N, then |s_{n}| < \epsilon. In particular, there exists N_{1} such that if \epsilon > 0 and  n > N_{1}, then |s_{n}| < \frac{\epsilon}{M}. Hence, |s_{n}t_{n}| = |s_{n}||t_{n}| < \frac{\epsilon}{M}t_{n} \le \frac{\epsilon}{M}M = \epsilon, given that \epsilon > 0 and n > N_{1}. And so, lim\,(s_{n}t_{n}) = 0. Q.E.D.

    2)Suppose that (s_{n}) and t_{n} are sequences such that |s_{n}| \le t_{n} for all n and lim\, t_{n} = 0. Prove lim\, s_{n} = 0.

    PROOF:

    Assume the given conditions above. This means there exists N such that, if \epsilon > 0 and n > N, then |t_{n}| < \epsilon. Hence, |s_{n}|\le t_{n} = |t_{n}| < \epsilon, and so there exists N such that, if \epsilon > 0 and  n > N, then |s_{n}| < \epsilon. And so, lim\, s_{n} = 0. Q.E.D.

    Any feedback would be appreciated, thanks.
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