 Uniform ergodic convergence for uniquely ergodic systems

For a uniquely ergodic continuous map $T$ of a compact metric space, every continuous $f$ satisfies $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to \int fd\mu$ uniformly in $x$. Otherwise choose increasingly long orbit empirical measures with a fixed discrepancy. Compactness provides a weak limit; the telescoping identity makes it invariant, so it must be the unique measure $\mu$, contradicting that discrepancy. Compactness and continuity are essential hypotheses.