I'm super rusty on my real analysis and I'm stuck on this easy question.
Show that for a semi-continuous function $ g: \mathbb{R}^n \rightarrow \mathbb{R}$, the set $\{x \in \mathbb{R}^n: g(x) \leq 0\}$ is closed
I have found some proofs online though I am working with somewhat different definitions of semi-continuous functions and closed sets and am trying to see whether sticking to the definitions I know.
In particular:
A closed set is a set such that all limits points are within the set.
A point $x \in S$ is a limit point of $S$ if $\forall \delta > 0, \exists y \in S$ such that $y \neq x$ and $|x-y| < \delta$.
A function $g$ is lower semicontinuous at $x$ if $\forall \epsilon >0, \space \exists \delta > 0$ such that $g(x) - g(y) < \epsilon \space \forall y \in N_{\delta}(x)$