consider a uniformly continuous $f:(a,b) to R$, prove $\lim$ at b exists

Question

I choose $x_n$ to be a cauchy sequence in (a,b). Then, for $n, m \geq N$, $|x_n - x_m| < \epsilon$. Since f is uniformly continuous, then $|x_n - x_m| < \epsilon$ gives f is cauchy. But I don't know how to go further.

Answer 1

• Choose $x_n$ to be a sequence converging to $b$.
• $(x_n)$ is therefore a Cauchy sequence
• Your work shows that $(f(x_n))$ is also a Cauchy sequence.
• Therefore...