$f_n: [0,1]\to \mathbb{R}$
$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.
Is it a Cauchy sequence?
2026-04-04 08:39:56.1775291996
Sequence in the complete metric space
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It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.