A Continuous Function on a Closed Interval is Riemann Integrable

Let $a,b$ be real numbers with $a<b$ in the order of the ordered field $\mathbb{R}$, let $[a,b]$ be the closed interval determined by $a$ and $b$, regarded as a subset of the real line $(\mathbb{R},d_{\mathbb{R}})$, and let the codomain $\mathbb{R}$ carry the same metric $d_{\mathbb{R}}$. Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$.

Then the following hold.

1. (Integrability) ¶ $f$ is Riemann integrable on $[a,b]$.

2. (Closed subintervals) ¶ For all $u,v\in[a,b]$ with $u<v$, the restriction $f|_{[u,v]}$ is Riemann integrable on $[u,v]$.