Proving every montonic function on an interval is integrable I am trying to understand the proof of every monotonic function that is on an interval is integrable.
This is what I have $U(f, P) - L(f, P) = \sum\limits_{k=1}^n(f(t_k) - f(t_{k-1}))\cdot (t_k - t_{k-1})$
Now my book says that this is equal to:
$= (f(b) - f(a))\cdot(t_k - t_{k-1})$
How does one deduce that $\sum\limits_{k=1}^n(f(t_k) - f(t_{k-1})) = f(b) - f(a)$?

You deduce that last identity because the sum telescopes. If you write 
$$
\sum_{k=1}^n (f(t_k) - f(t_{k-1})) = \sum_{k=1}^n f(t_k) - \sum_{k=0}^{n-1} f(t_k) = f(t_n) - f(t_0), 
$$
if $t_n = b$ and $t_0 = a$, this is what you want. The function must be integrable when $f$ is monotonic because
$$
\left| \sum_{k=1}^n (f(t_k) - f(t_{k-1}))(t_k - t_{k-1}) \right| \le \max \{ (t_k - t_{k-1}) \} \sum_{k=1}^n |f(t_k) - f(t_{k-1})|.
$$
By assumption on f monotonic, the absolute values get removed and all you keep is the sign of $f$ (i.e. + if $f$ increases, $-$ if it decreases). Therefore that sum telescopes to $f(b) - f(a)$, i.e. a constant that does not depend on the partition. Since the step goes to $0$ (the maximum in front of the sum), $f$ is integrable.
Hope that helps,
