Riemann integral/Step functions with equal limit/Integral/Fact

Let ${}I$ denote a compact interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. Suppose that there exists a sequence of lower step functions ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ with ${}s_{n}\leq f$ and a sequence of upper step functions ${}{\left(t_{n}\right)}_{n\in \mathbb {N} }$ with ${}t_{n}\geq f$. Suppose furthermore that the corresponding sequences of step integrals converge to the same real number.

Then ${}f$ is

Riemann-integrable, and the definite integral equals this limit, so

${}\lim _{n\rightarrow \infty }\int _{a}^{b}s_{n}(x)\,dx=\int _{a}^{b}f(x)\,dx=\lim _{n\rightarrow \infty }\int _{a}^{b}t_{n}(x)\,dx\,.$

Write a proof