Proof of Riemann Integrability Criterion via Upper and Lower Sums

Suppose $f:[a,b]\to\mathbb R$ is bounded. Then $f$ is Riemann integrable iff for every $\varepsilon>0$ there exists a partition $P$ with $U(f,P)-L(f,P)<\varepsilon$. If this holds, refining partitions produces upper sums decreasing and lower sums increasing with arbitrarily small gap, so the upper and lower integrals coincide. Conversely, if $f$ is integrable, upper and lower integrals are equal to the common integral value, hence there exists a partition with gap $<\varepsilon$ by the definitions of infimum/supremum of upper/lower sums.