Algebra of Riemann integrable functions

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

Proposition:

• For any $\alpha,\beta\in\mathbb{R}$, the function $\alpha f+\beta g$ is Riemann integrable.
• The product $fg$ is Riemann integrable.
• Consequently, $f^2$ is Riemann integrable.

These closure properties are essential: they guarantee the Riemann integral behaves well under the usual algebraic operations on functions.