I know that a sequence in $\mathbb{R}^d$ is Cauchy iff it is convergent, and also convergent sequences are Cauchy in a general metric space. How does the converse fail in a general metric space? In other words, what part of the proof that Cauchy sequences in $\mathbb{R}^d$ are convergent cannot be generalized to an arbitrary metric space?
2026-03-29 15:59:53.1774799993
Cauchy sequence nonconvergent in general metric space
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The sequence $$\left\{\frac1n\right\}_{n=1}^\infty$$ is Cauchy, but not convergent, in $(0,\infty)$ equipped with the standard metric.