Weierstrass Thm: Continuous Fn Attaining Extrema on Compact Domain $u:\mathbb{R^L}\rightarrow\mathbb{R}$
The Weierstrass Extreme Value Theorem guarantees this. However, it's used so frequently that economists take it for granted or omit it in graduate courses and research papers. MWG is a maturity book. Part of mathematical maturity is being able to fill in the details for a proof.