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

Let ${}I=[a,b]$ be a compact interval and let

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

be a function. Consider a sequence of staircase functions ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ such that ${}s_{n}\leq f$ and a sequence of staircase functions ${}{\left(t_{n}\right)}_{n\in \mathbb {N} }$ such that ${}t_{n}\geq f$. Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that ${}f$ is Riemann-integrable and that

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

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