lower semicontinuity and closed graph

Question

I would like to prove that for a LSC function, its epigraph is closed. I saw some longer proof here, but why would the following not hold ? :

$f LSC := \liminf f(x_n) \ge f(x)$ when $x_n \rightarrow x$

then $x_n,a_n \rightarrow x,a \implies f(x) \le \liminf f(x_n) \le \liminf a_n = a$

so $x,a \in epi(f)$

Answer 1

This is a streamlined version of the proof in wikipedia. Perhaps it's a bit too streamlined (write it up using quantifiers!). Otherwise it appears perfectly valid to me.