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 $\sum\limits_{k=1}^n(f(t_k) - f(t_{k-1})) = f(b) - f(a)$?

Note that
\begin{equation*}
\sum_{k=1}^{n}(f(t_{k})-f(t_{k-1}))=(f(t_{1})-f(t_{0}))+(f(t_{2})-f(t_{1}))+(f(t_{3})-f(t_{2}))+...+(f(t_{n})-f(t_{n-1})),
\end{equation*}
so in $i$:th term of the sum, $-f(t_{i-1})$ always eliminates the $i-1$:th appearing term $+f(t_{i-1})$. Hence you are only left the endpoints, i.e. $f(b)-f(a)$.
