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I know that Cauchy sequences in metric spaces can fail to converge (but if the metric space is complete, they always converge). Can there be a convergent sequence (perhaps in a complete metric space) that is not Cauchy, or does "Cauchyness" follow from convergence in a complete space?

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Suppose $\lim_{n \to \infty} x_n =x.$ Then $$d(x_n,x_m)\leq d(x_n,x)+d(x,x_m)\to0 \text{ as }n,m\to\infty$$ Hence, every convergent sequence in any metric space is Cauchy.

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