While reading Bartle's book, i came across the following:
Problem:
Prove that if $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R},$ then $f$ is bounded on $A$.
Attempt:
Given $A \subset \mathbb{R}$ such that $A$ is bounded and $f: A \rightarrow \mathbb{R}$ an uniformly continuous function, we will prove that $f$ is bounded.
Given $(x_{n})$ a sequence of points in $A$ one can obtain a convergent subsequence $(x_{nk})$, by using the Bolzano-Weierstrass theorem. Since $f$ is uniformly continuous and $(x_{nk})$ is a Cauchy sequence (since it is convergent), $(f(x_{nk}))$ is also Cauchy.
Finally, since every Cauchy sequence is bounded, there exists $a$ and $b$ in $\mathbb{R}$ such that: $$a \leqslant f\left(x_{n k}\right) \leqslant b$$
Therefore, $f$ is bounded.
Questions:
1. Is the solution correct?
2. If the solution is indeed correct, is it well written?
3. If the solution is incorrect, can someone gently me explain why and provide a solution?
Thanks in advance, Lucas!