Math 742 : Geometric Analysis Lecture 5 and 6 Notes

Proof. If E ≡ +∞ onX, then there is nothing to prove. So, assume E 6≡ +∞. Denote α = infu∈X E(u) ≥ −∞ and choose a strictly decreasing sequence (αm) such that αm ↘ α as m → ∞. Let Kαm = {u ∈ X : E(u) ≤ αm}, then by assumption Kαm is compact. Moreover, it’s clear that Kαm+1 ⊂ Kαm for all m ∈ N, i.e. we have a decreasing nesting of compact sets. By Cantor’s Intersection Theorem, we know that K = ∞ ⋂