Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step functions converging uniformly to $f$. Explain why such a sequence always exists and give a proof that the limit does not depend on the choice of sequence.
There is always a sequence of step functions that converge uniformly to $f$ because $f$ is regulated and this is precisely the definition of a regulated function.
How would I prove that the limit doesn't depend on the choice of sequence?
Thanks in advance.
If $\{s_{n}\}_{n\in\mathbb{N}}$ and $\{t_{n}\}_{n\in\mathbb{N}}$ are two sequences of step functions that both converge uniformly to $f$ then $s_{n}-t_{n}$ converges uniformly to $0$ and so $$\int_{a}^{b}s_{n}\,dx-\int_{a}^{b}t_{n}\,dx\to0$$ as $n$ tends to infinity.