Show that $f(b) - f(a) \geq \sum_{n=1}^{\infty}(f(b_n)-f(a_n))$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an increasing function and suppose that:

• $A = \bigcup_{n=1}^\infty(a_n,b_n]\ \ (\text{ disjoint union }) \ \ (a_n \leq b_n; ,a_n ,b_n \in [-\infty, \infty])$;

• $A\subseteq [a,b] \ \ (a<b; a,b \in [-\infty,\infty])$

Denote $f(\infty) = \sup f, f(-\infty) = \inf f$

I need to show that $f(b) - f(a) \geq \sum_{n=1}^{\infty}(f(b_n)-f(a_n))$ for my homework. My idea was to show that $f(b) - f(a) \geq \sum_{n= 1}^m(f(b_n)-f(a_n)),\ \forall m \geq 1$. But I still don't know how to proceed. Could anyone help me?