Let $\| \cdot \|_0$ be $0$-norm on $\mathbb{R}^n$ defined as $$ \|\cdot \|_0 := \text{the number of nonzero elements in}\,\,x, \forall x\in \mathbb{R}^n $$ Show that $\| \cdot \|_0$ is lower semi-continuous on $\mathbb{R}^n$.
We can use the original definition of lower semi-contnuity as
Def: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is lower semi-continuous at $x_*$ if $\forall \epsilon >0$ there exist $\delta >0 $ such that $\forall x$ with $\|x-x_*\|< \delta$,
$$f(x_*)-\epsilon \leq f(x)$$
Or we could show it using the following Lemma:
Lemma: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is lower semi-continuous at $x_*$ if and only if for any sequence $(x_n) \rightarrow x_*$ : $f(x_*) \leq \lim \inf f(x_n)$.
I think to show it we need to use the above Lemma.
What happens if you take $n=1$ and consider $x_n = 1/n$?
$1 = \lim \inf ||1/n||_0 > f(0) = 0$