let $f:\mathbb{R\to R}$ bounded and continuos, and let (for $n\in\mathbb{N}$) $f_{n}(x)=\sup\{f(y):y\in[x-\frac{1}{n},x+\frac{1}{n}]\}.$
prove that $f_{n}\rightrightarrows$ f on every proper and bounded interval on $\mathbb{R}$
so long I have tried using Dini's theorem. of course that $f_{n}$ is monotonically increasing, and it is easy to prove the pointwise convergence, but I cant grasp the idea of proving that every (or almost every $f_{n}$) is continuous.
thank you!