In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact.
The function is $u_n(x):= (1/n)G(nu(x))$, where $G \in C^1(\mathbb{R})$ and verifies $G(t) \leq |t|$ and: $$ G(t)=\left\{\begin{matrix} 0 & if \ |t| \leq 1\\ t & if \ |t| \geq 2 \end{matrix}\right.$$
And $u$ belongs to the Sobolev Space $W^{1,p}(I)$, where $I$ is a (possibly unbounded) interval in $\mathbb{R}$. Also, $u=0$ on $\partial(I)$. Recall that if $I$ is unbounded, then it verifies $$\lim_{x\in I, |x| \to \infty}{u(x)}=0 \quad \quad (1)$$.
What the proof says is:
$supp(u_n) \subset \{x \in I \ | \ |u(x)| \geq 1/n \}$, so, using the fact that $u=0$ on $\partial(I)$ and (1) then $supp(u_n)$ is compact.
And this is what I don't understand. Where is the marked in bold used? As far as I know, checking if a set is compact in $\mathbb{R}$ can be done by checking if it is closed (Which it is because the definition of the support) and bounded, which it is because it is a subset of a bounded set.
And the last fact is due to (1) if $I$ is unbounded and there is no assumption needed if $I$ is bounded. What am I missing?