I am self-learning Real-Analysis. I'd like a clue or hint on how to proceed with this exercise problem from Stephen Abbott's Understanding Analysis without revealing the entire proof.
[Abbott, 6.2.10] This exercise and the next explore partial converses of the Continuous Limit Theorem (Theorem 6.2.6). Assume that $f_n \to f$ pointwise on $[a,b]$ and the limit function $f$ is continuous on $[a,b]$. If each $f_n$ is increasing (but not necessarily continuous), show that $f_n \to f$ uniformly.
Proof Attempt.
I tried a direct proof, but I am not exactly sure how to use the fact that each $f_n$ is increasing.
I tried a proof with contradiction.
We are given that, $f_n \to f$ pointwise on $[a,b]$ and the limit function $f$ is continuous on $[a,b]$. Further each $f_n$ is increasing. We proceed by contradiction.
Assume that $f_n$ does not converge uniformly to $f$. Thus,
\begin{align*} (\exists \epsilon_0 > 0), (\forall N \in \mathbf{N}) : (\exists n_0 \geq N),(\exists x_0 \in [a,b]) : |f_n(x_0) - f(x_0)| \geq \epsilon_0 \end{align*}
Since, $f$ is continuous on the compact set $[a,b]$, it is uniformly continuous on $[a,b]$. Now, I am not sure, how to proceed.
Any hints would be super-helpful guys.