If $f: (a,b)\to\mathbb R$ is uniformly continuous and $\{x_n\}$ in the domain tends to $b$, then why does $\{f(x_n)\}$ have a limit?
2026-04-02 18:39:54.1775155194
The limit at the end-point of an open interval, for a uniformly continuous function
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Because $x_n$ is Cauchy sequence then $f(x_n)$ is Cauchy sequence due to uniform continuouity. And hence $f(x_n)$ is convergant.