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

Put $s_i={\sup }_{C_i}f$ and omit empty sets. Since the sets cover the space, boundedness givesFor nonnegative numbers $u_{n,i}$, their sum lies between their maximum and m times their maximum. Consequently its scaled logarithmic upper limit is at most ${\max }_i{\mathrm{limsup}}_n{n}^{-1}\log u_{n,i}$; the added $n^{-1}\log m$ vanishes. Each $C_i$ is closed, so the upper bound in the large deviation principle givesThe convention $\log 0=-\infty$ also handles sets of zero probability or infinite infimum rate.