If f is monotone on a closed interval then it is absolutely continuous

Question

How do I prove that if a function f is monotone on a closed interval then it is absolutely continuous, I know that if it is monotone that it has bounded variation, but that doesn't imply that $$\sum_{i=1}^n |f(x_{i+1})-f(x_i))|<\epsilon$$ for all $\epsilon>0$ it only implies that this sum is finite.