Let $T\subset T'$ such that $T'$ contains $p$ more points than $T$. Then the following is true:
$0\le U(f,T)-U(f,T')\le p \cdot (M-m)\cdot h(T)$
$0\le L(f,T')-L(f,T)\le p \cdot (M-m)\cdot h(T)$
In this case $m=$inf{$f(x)|x\in [a,b]$} and $M=$sup{$f(x)|x\in [a,b]$}
Also if anyone can tell me the name of this theorem or how to find it I would appreciate it. Spent the last hour digging through calculus and analysis books but nothing that resembles it.