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Math Help - Proving convergent seq. is a cauchy seq.

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    Proving convergent seq. is a cauchy seq.

    I proved convergent sequences and cauchy sequences.
    How would I prove that every convergent sequence is a Cauchy sequence?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by summerset353 View Post
    I proved convergent sequences and cauchy sequences.
    How would I prove that every convergent sequence is a Cauchy sequence?
    Let \varepsilon>0 be given. There exists some N\in\mathbb{N} such that N\leqslant n\implies d(x_n,x)<\frac{\varepsilon}{2} and so N\leqslant m,n\implies d(x_n,x_m)\leqslant d(x_n,x)+d(x_m,x)=\varepsilon
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