I am trying to prove the following:
Show that an increasing, continuous function $f$ on $[a,b]$ is integrable there.
This is my idea:
Let $f$ be an increasing, continuous function on $[a,b]$. Take a partition $P$ of $[a,b]$ of $n$ equal-length sub-intervals. Then $\begin{align} U(f,P)-L(f,P)&=\sum_{k=1}^{n}(M_k-m_k)(x_k-x_{k-1})=\sum_{k=1}^{n}[f(x_k)-f(x_{k-1})]\Delta x\\&=[f(a)-f(b)]\Delta x. \end{align}$ Since $\lim_{n\to\infty}\Delta x=0$, it follows that $U(f,P)-L(f,P)\to0$. Hence, $f$ is integrable.
Does this seem reasonable? I believe I heard that $\epsilon$ definitions were necessary for this.