Showing that all monotone functions are integrable
Solution 1:
$$\begin{align}\sum_{i=1}^n[f(x_i)-f(x_{i-1})] &= \sum_{i=1}^n f(x_i)-\sum_{i=1}^nf(x_{i-1}) \\&= \sum_{i=1}^n f(x_i)-\sum_{i=0}^{n-1}f(x_{i})\\&=f(x_n)-f(x_0)\end{align}$$
Solution 2:
Just try to write down that sum.
You'll see that each term cancels out except the first and the last, leaving you with that expression.
By the way, it is false that if $f(x)$ is monotone on $[a, b)$ then $f(x)$ is bounded.. just take a function with an vertical asymptote.