Proof of Continuous Functions on Compact Intervals are Riemann Integrable

Let $\varepsilon>0$. By Heine-Cantor: Continuity on Compact Interval Implies Uniform Continuity, there exists $\delta>0$ such that $|x-y|<\delta$ implies $|f(x)-f(y)|<\varepsilon/(b-a)$. Choose a partition $P$ with mesh $|P|<\delta$. On each subinterval $I_i=[x_{i-1},x_i]$, the oscillation satisfies $\omega_i:=\sup_{I_i}f-\inf_{I_i}f<\varepsilon/(b-a)$. Hence $U(f,P)-L(f,P)=\sum_i \omega_i\,|I_i| < \frac{\varepsilon}{b-a}\sum_i |I_i| = \varepsilon.$ By Riemann Integrability Criterion via Upper and Lower Sums, $f$ is Riemann integrable on $[a,b]$.