The limit at the end-point of an open interval, for a uniformly continuous function 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?

Because $x_n$ is Cauchy sequence then $f(x_n)$ is Cauchy sequence due to uniform continuouity. And hence $f(x_n)$ is convergant.