Let $f_n \rightarrow f$ pointwise on $D$, and let $f_n$ be monotone increasing on $D$; prove that $f$ is monotone increasing on $D$.

Question

Consider the following problem from Dangello and Seyfreid:

If $f_n \rightarrow F$ pointwise on $D$, and each $f_n$ is monotone increasing on $D$, show that $F$ is monotone increasing on $D$.

How can I prove this? I think the result has something to do with $f_n$ converging uniformly to F?

Answer 1

Since each $f_n$ is monotone increasing you have that for any $x_1>x_2 \in D$ \begin{align}f_n(x_1)\ge f_n(x_2)\;\; \forall n \in \Bbb N &\implies \lim_{n\to+\infty}f_n(x_1)\ge \lim_{n\to+\infty}f_n(x_2)\\[0.2cm]&\implies F(x_1)=\lim_{n\to+\infty}f_n(x_1)\ge \lim_{n\to+\infty}f_n(x_2)=F(x_2)\end{align} where is the second implication holds by the pointwise convergence.