Theorem 1   Continuous functions defined on compact subsets of $\R^n$ are Riemann integrable.

Proof. Let $D\subset \R^n$ be a compact subset of $\R^n$ and let $f\colon D\to \R$ be a continuous function. Since $f$ is defined on a compact set, $f$ is uniformly continuous i.e. given $\epsilon>0$ there exists $\delta>0$ such that $|x-y|\le \delta \Rightarrow |f(x)-f(y)|\le \epsilon$ Let $R>0$ be large enough so that $D\subset (-R,R)^n$ (such an $R$ exists because $D$ is bounded). Let $P$ be a polyrectangle such that $D\subset \cup P \subset (-R,R)^n$ and such that every rectangle $R$ in $P$ has diameter which is less then $\delta$ So one has $\sup_R f(x)-\inf_R f(x) \le \epsilon$ and hence $$ S^*(f,P)-S_*(f,P)\le \epsilon \sum_{Q\in P} \mathrm{meas} (Q) \le \epsilon \mathrm{meas}(P) \le \epsilon \mathrm{meas}[-R,R]^n = \epsilon 2^n R^n. $$ Letting $\epsilon\to 0$ one concludes that $S^*(f)=S_*(f)$