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

Let $b\_k \subset [a,b]$ be a sequence such that $\lim\_{k\rightarrow \infty} {b\_k} = b$

Consider the integral $\int\_a^{b}f(x) dx$.
Is it always the case that $$\int\_a^{b}f(x) dx = \lim\_{k\rightarrow \infty } \int\_a^{b\_k}f(x)?$$

If yes, why? If no, could you please provide a counterexample?

I feel that is the case, however, not sure how to prove it(or at least start my proof)