Word Count : 256

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No, the continuous image of a Cauchy sequence need not be a Cauchy sequence in general. A sequence ((x_n)) in a metric space ((X, d_X)) is Cauchy if for every (\varepsilon > 0) there exists (N) such that for all (m, n \ge N), (d_X(x_m, x_n) < \varepsilon). Suppose (f : X \to Y) is continuous and ((x_n)) is Cauchy in (X). Then ((f(x_n))) ________ _______ ___ ______ ____.
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