Suppose a sequence of $f_n$ is non-decreasing convergent to $f$( $f_n\geq f_{n-1}$). Is the epigraph formed by $f_n$ also convergence to that formed by $f$?
For instance $S_n = \{(x,y):f_n(x)\leq y\},S = \{(x,y):f(x)\leq y\}$.$\lim_{n\rightarrow \infty}S_n=S$?
We know that $\lim_{n\rightarrow \infty}S_n = \bigcap_{n\geq 1} S_n$. But I am not quite sure whether $\bigcap_{n\geq 1} S_n=S$ is true