If $f : [a, \infty) \rightarrow \Bbb R$ is monotonically decreasing and the integral $\int_0^\infty f(x) \,dx$ is convergent, then $\lim_{x\rightarrow\infty} f(x) = 0$. I don't really understand why this is true.

I know that if $f$ is monotonically decreasing then $f$ is bounded ($\exists M\in \Bbb R s.t. f(x)\leq M , \forall n\geq k)$.

I also know that $$\int_0^\infty f(x) \,dx = \lim_{b\rightarrow\infty}\int_0^b f(x) \,dx < \infty$$ I guess I can say something about the the monotonicity of limits, but I don't know how to continue from here. I would love for some help (it doesn't need to be formal proof, I just want to understand it).

Thank you.