How do I show that a set is closed, bounded and convex?

Question

How do I show that the set
{$u\in L^2(\Omega):u_a(x) \leq u(x) \leq u_b(x) for \space a.e \space \space x \in\Omega$}
is convex and closed.

How would you use the following result to prove it?:

if $\mid {u_n}\mid_{L^2(\Omega)} \rightarrow0$ then the subsequence of {$u_n$}$_{n=1}^\infty$ that converges to $0$ a.e in $\Omega$.

Answer 1

Convexity is straightforward by using the definition.

Then, if $u_n$ is a sequence in your set, which converges towards $\bar u$, you can use the hint to get $u_n(x) \to \bar u(x)$ for a.a. $x$ along a subsequence. This can be used to deduce that $\bar u$ also belongs to your set.