Let $[a,b]$ be an interval in $\mathbb{R}$. Let $P=\{x_0,...,x_n\}$ be a partition of $[a,b]$ and $f$ be a real function bounded on $[a,b]$. Let $\alpha$ be a monotonically increasing function on $[a,b]$
Suppose $\forall s\in \Pi_{1≦i≦n} [x_{i-1},x_i], \sum_{i=1}^n f(s_i)[\alpha(x_i) - \alpha(x_{i-1})]< A$ for some $A\in \mathbb{R}$.
I don't know how to prove that $U(P,f,\alpha)≦A$.
Help.. Thank you in advance.