I'm working on a question on uniform convergence. After spending an hour, I'm still trying to write a formal proof for it. The question seems obvious to me, but I cannot prove it formally.
Consider $f_n$ as a sequence of continuos functions pn [a,b]. Suppose for any $x \in [a,b]$ $f_n(x)$ is non-increasing. I need to show that if $f_n \rightarrow 0$ pointwise pn [a,b], then $f_n \rightarrow 0$ uniformly on [a,b].
Thank you very much for your help.
This is what I've done so far:
I tried to prove it with contradiction. Let's suppose that $f_n$ does not converge to 0 uniformly, so then there exists a sequence $x_n \in [a,b]$ that converges to $x_0$ which is for sure in [a,b] and for that series, we have an epsilon for which $f_n(x_n) \ge 0$. I don't know how to continue from here. Can I say since f converges point wise, so then for every element of $x_n$ (let's say $x_k$), then $f_n(x_k)$ converges to 0 as n goes to $\infty$ and therefore the limit should converge to?