This Theorem is from the book Measure and Integral by Zygmund & Wheeden:



According to this given $\epsilon\gt 0$ there exist a $\delta\gt 0$ such that for any partition $\Gamma$, if $|\Gamma|\lt\delta$, then $U_\Gamma-L_\Gamma\lt\epsilon.$
So, if your $f$ is bounded (it must be, otherwise the $U(P,f,\alpha)$ or $L(P,f,\alpha)$ might have no sense), given $\epsilon\gt 0$, in order to pick a uniform partition $P=\{a=x_0\lt\cdots\lt x_n=b\}$ such that $U(P,f,\alpha)-L(P,f,\alpha)\lt\epsilon,$ it is enough to choose $n$ large enough so that $\frac{b-a}{n}\lt\delta.$