Discontinuities on a Bounded Increasing Function

Question

I'm working on question 2.35 from Carothers and I'm not sure how to get started. Here it is:

Let $f:[a,b] \to R$ be increasing and let {$x_n$} be an enumeration of the discontinuities of $f$. For each $n$, let $a_n=f(x_n)-f(x_n-)$ and $b_n=f(x_n+)-f(x_n)$ be the left and right "jumps" in the graph of $f$, where $a_n=0$ if $x_n=a$ and $b_n=0$ if $x_n=b$. Show that $\sum_{n=1}^\infty a_n \le f(b)-f(a)$ and $\sum_{n=1}^\infty b_n \le f(b)-f(a)$.

A few things I noticed:

1. Since $f$ is increasing, on $[a,b]$, we have $f(x) \le f(y)$ for all $x,y \in [a,b]$ where $x \le y$.
2. For any $x \in [a,b]$, $f(x-)$ and $f(x+)$ both exist.
3. Furthermore, we can show that $f$ has countably many jump discontinuities, but that does not seem helpful here.

Any pointers on how to proceed would be great. Thank you-

Answer 1

By reductio ad absurdum.

If $\sum_{n=1}^\infty a_n > f(b)-f(a)$ then you have a finite sequence $a0.$$

By definition of the limit, you can find $$a < y_1 < x_1 0.$$

In contradiction with $$(f(x_1)-f(y_1)) + (f(x_2)-f(y_2)) + \dots + (f(x_m)-f(y_m)) \le f(b)-f(a)$$