Riemann integrability implies boundedness

Proposition: If $f:[a,b]\to\mathbb{R}$ is Riemann integrable on $[a,b]$, then $f$ is bounded on $[a,b]$.

In many texts, boundedness is built into the definition of Riemann integrability. This proposition records that boundedness is not optional: unbounded functions cannot have finite upper and lower sums .

Proof sketch: If $f$ were unbounded above, then for every partition $P$ there would be some subinterval on which $\sup f=+\infty$, forcing $U(f,P)=+\infty$. Similarly if unbounded below, some lower sum would be $-\infty$. Thus the equality of upper and lower integrals (and finiteness of the integral) cannot hold unless $f$ is bounded.