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Math Help - Convergent sequence

  1. #1
    zeh
    zeh is offline
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    Convergent sequence

    Let
    \lim\limits_{n\to\infty}q_n=q
    \lim\limits_{n\to\infty}p_n=p
    Prove that
    \lim\limits_{n\to\infty}min\{q_n,p_n\}=min\{q,p\}
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  2. #2
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    Quote Originally Posted by zeh View Post
    Let
    \lim\limits_{n\to\infty}q_n=q
    \lim\limits_{n\to\infty}p_n=p
    Prove that
    \lim\limits_{n\to\infty}min\{q_n,p_n\}=min\{q,p\}
    Suppose q = p. Then every subsequence of q and p converge to q = min{q, p}, hence the minimum of q_n and p_n converges to q. On the other hand, suppose without loss of generality that q < p. Let r = (p + q)/2. Then there exists an N such that for all n >= N, |q - q_n| < r - q. Also, there exists an M such that for all m >= M, |p - p_n| < p - r. Let K = max{M, N}. Then q_k < (p + q)/2 < p_k for all k >= K, hence min{p_k, q_k} = q_k for all k >= K. Therefore the sequence converges to q.
    Last edited by rn443; October 26th 2009 at 07:40 AM.
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