Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 1 / d / Solution

Let epsilon decrease to zero in the upper estimate. In the lower estimate first take the supremum over all chosen points and then decrease epsilon to zero. The bounds coincide, proving Varadhan lemma:The value is finite: f is bounded, $I\ge 0$, and the large-deviation bounds on the whole space force $\inf I=0$. This proof actually uses bounded continuity and the large deviation principle; compact sublevels are not additionally needed for this bounded version.