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    Cauchy

    Prove that the sum of two Cauchy sequences is Cauchy without using:

    Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges.
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    Quote Originally Posted by taypez View Post
    Prove that the sum of two Cauchy sequences is Cauchy without using:

    Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges.
    define \{ z_n \} = \{ x_n \} + \{ y_n \}. we need to show that: for all \epsilon > 0, there exists a N \in \mathbb {N}, such that m,n > N implies |z_n - z_m| < \epsilon

    hint: you will need the triangle inequality here

    can you continue?
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    That would lead me to use the Cauchy Theorem which it said not to and since the Cauchy Thm uses Bolzano-Weierstrass, I'm assuming I can't use that either.
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    Quote Originally Posted by taypez View Post
    That would lead me to use the Cauchy Theorem which it said not to and since the Cauchy Thm uses Bolzano-Weierstrass, I'm assuming I can't use that either.
    no it would not. did you try it? all you need is the definition of Cauchy and the triangle inequality, nothing else. and that is allowed for this question
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    Quote Originally Posted by taypez View Post
    Prove that the sum of two Cauchy sequences is Cauchy without using:

    Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges.
    It is a trivial problem.

    |x_n+y_n - x_m - y_m|\leq |x_n-x_m|+|y_n-y_m| < \frac{\epsilon}{2}+\frac{\epsilon}{2} = \epsilon.
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    Bored.

    Let \{ x_n \} and \{ y_n \} be Cauchy sequences.

    Define \{ z_n \} = \{ x_n \} + \{ y_n \}. We show that \{ z_n \} is Cauchy.

    Since \{ x_n \} is Cauchy, for all \epsilon > 0, there exists an N_1 \in \mathbb {N} such that m,n > N_1 implies |x_n - x_m|< \frac {\epsilon}2.

    Similarly, since \{ y_n \} is Cauchy, we can find an N_2 \in \mathbb{N} such that m,n > N_2 implies |y_n - y_m|< \frac {\epsilon}2

    Now, fix such an \epsilon > 0, and choose N = \mbox {max} \{ N_1, N_2 \}. Then m,n > N implies that:

    |z_n - z_m| = |(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

    Thus, \{ z_n \} is a Cauchy sequence

    QED.
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