Let $f_n: [a, b] \to \Bbb R$ be continuous on $[a, b]$, and construct a monotone decreasing sequence of functions $(f_n)=(f_1, f_2, ...)$ converging pointwise to $f(x)=0$. Show that $f_n \to 0$ uniformly.
I've seen several different approaches to the problem posted around the internet, but wanted to play around with it a little. Would it be appropriate to consider the supremum norm here?
My initial intuition suggests that a strictly decreasing sequence of functions $(f_n)$ would surely have $||f_n(x)||_\infty > ||f_{n+1}(x)||_\infty$. In this case, I presume one would like to show that $||f_n(x)||_\infty \to 0$ and hence the convergence is uniform, but am not exactly sure how to go about this.
*This happens to be my first ever post, so if it leaves anything to be desired, please let me know
This approach can be made to work. Indeed by continuity of the $f_n$, for each $n\ge 1$ there exists $x_n \in [a,b]$ such that $f_n(x_n) = ||f_n||_\infty$. It suffices to show that $||f_{n_k}||_\infty \to 0$ for some subsequence $f_{n_k}$ (since $||f_n||_\infty$ is decreasing), so by passing to a suitable subsequence of $(f_n)$ we may assume $x_n \to c$ for some $c \in [a,b]$ (compactness of $[a,b]$).
Now we consider the behaviour of these functions near $c$. Fix some $\epsilon>0$. Since $f_n \to 0$ (monotonically) pointwise, there is $N\ge 1$ so that $f_n(c) < \epsilon$ for all $n\ge N$. Furthermore, since $f_N$ is continuous we can choose $\delta>0$ such that $f_N(x) < 2\epsilon$ for $x \in (c-\delta,c+\delta)\cap [a,b]$.
Since $x_n \to c$, for all $n$ sufficiently large we have $x_n \in (c-\delta,c+\delta)\cap [a,b]$, implying $$||f_n||_\infty = f_n(x_n) \le f_N(x_n) < 2\epsilon$$ for all $n$ sufficiently large. Since $\epsilon>0$ was arbitrary we conclude $||f_n||_\infty \to 0$ as required.