Let $f_{n}:M\to \mathbb{R}$ a sequences of continuous function that converges pointwise to continuous function $f:M\to\mathbb{R}$. Then for each $n\in\mathbb{N}$, given $\epsilon>0$, define the set $F_{n}=\{x\in M;\vert f_{n}(x)-f(x)\vert\geq\epsilon\}$. Show that each $F_{n}$ is closed.
Maybe is more easy show that $\{F_{n}\}^{c}$ is open. Any hint plis.
My question is because, I wanna show the Dini's theorem: If a sequences of real-valued functions $f_{n}:M\to\mathbb{R}$, defines in a compact metric space M, converges pointwise to continous function $f:M\to\mathbb{R}$, and furthermore, we have $f_{n}(x)\leq f_{n+1}(x)$, for all $x\in M$, then the convergency $f_{n}\to f$ is uniformly in M.
The answer in my book says: Given $\epsilon>0$, define, for each $n\in\mathbb{N}$, $F_{n}=\{x\in M;\vert f_{n}(x)-f(x)\vert\geq\epsilon\}$. Then $F_{1}\supset F_{2}\supset ...\supset F_{n}\supset ...$ and each $F_{n}$ is closed in M. But I don't see that each $F{n}$ is closed. Regards!
