general topology

general topology

Let $f:X\to \mathbb{R}$ be a map where $X$ is a topological space.The map
$f$ is said to be lower semi continuous if for any $a\in \mathbb{R}$,
$f^{-1}((-\infty,a])$ is closed in $X$. Prove that if there is a sequence
of lower semi continuous functions $f_n$ and $f=\sup f_n$ then $f$ is
lower semi continuous.