Absolute value preserves Riemann integrability

Let $f:[a,b]\to\mathbb{R}$ be Riemann integrable .

Proposition: The function $|f|$ is Riemann integrable on $[a,b]$. Moreover, $\left|\int_a^b f(x)\,dx\right|\le \int_a^b |f(x)|\,dx.$

This is a basic stability property: composing with the continuous map $x\mapsto |x|$ preserves Riemann integrability.