I am reading this book in analysis. I want to check if my proof is correct.
Let $a < b$ be real numbers, and $f : [a,b] \rightarrow \mathbb{R}$ be a function which is both continuous and strictly monotone increasing. Then $f$ is a bijection from $[a,b]$ to $[f(a),f(b)]$, and the inverse $f^{-1} : [f(a),f(b)] \rightarrow [a,b]$ is continuous and strictly monotone.
Attempt:
Assume $f^{-1}$ isn't continuous. Then there exists a sequence $(y_n)_{n}$ such that $y_n$ converges to $y_0$, but $x_n = f^{-1}(y_n)$ doesn't converge to $x_0 = f^{-1}(y_0)$. By Heine Borel we have that the sequence $x_n$ has convergent subsequence $x_{n_j}$ that converges to $x_0^{\prime}$. By uniqueness of limits and continuity we must have $f(x_0^{\prime}) = y_0$. $f(x_0^{\prime}) = y_0 = f(x_0)$. It follows that $x_0^{\prime} = x_0$. I am not sure what to do from here.
Edit: I think it is from construction of the $x_i$ sequence, that all of its subsequence are convergent to $x_0$.
I think my proof works though Heine-Borel is heavy machinery to use here. I wonder if you can get without Heine-Borel