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Math Help - More Cauchy Sequences

  1. #1
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    More Cauchy Sequences

    If (Xsubn) and (Ysubn) are Cauchy Sequences, show that (Xsubn + Ysubn) and (Xsubn*Ysubn) are Cauchy Sequences. I am asked to prove this using the definition of a Cauchy Sequence only. Any help is much appreciated.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jkru View Post
    If (Xsubn) and (Ysubn) are Cauchy Sequences, show that (Xsubn + Ysubn) and (Xsubn*Ysubn) are Cauchy Sequences. I am asked to prove this using the definition of a Cauchy Sequence only. Any help is much appreciated.
    i will do the first (since it is easier ) you do the second

    since \{ x_n \} is a Cauchy sequence, we have that for all \epsilon > 0 there exists N_1 \in \mathbb{N}, such that n,m > N_1 implies |x_n - x_m| < \frac {\epsilon}2

    similarly, we have N_2 \in \mathbb{N} so that n,m > N_2 implies |y_n - y_m| < \frac {\epsilon}2

    Now, choose N = \text{max} \{ N_1,N_2 \}. then n,m > N implies

    |(x_n + y_n) - (x_m + y_m)| = |(x_n - x_m) + (y_n - y_m)| \le |x_n - x_m| + |y_n - y_m| < \frac {\epsilon}2 + \frac {\epsilon}2 = \epsilon

    so that \{ x_n + y_n \} is Cauchy


    (note that the \le follows by the triangle inequality)


    now do the second. use the defintion as i did
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