Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed
The statement does not seem to be true.

Define $u$ as $u(x) = -1$ if $x \leq 0$, $u(x) = 0$ if $x \in (0,1)$ and $u(x) = 1$ if $x \geq 1$. The set of points $x$ for which $u(x) \geq 0$ is $(0,\infty)$, which is not closed.