# Prove that if $\int_{a}^{b} f(x)$ exists, $\delta >0$ such that $|\sigma_1 -\sigma_2|<\epsilon$

Prove if $\int_{a}^{b} f(x)$ exists, then for every $\epsilon>0$, there is a $\delta >0$ such that $|\sigma_1 -\sigma_2|<\epsilon$, if $\sigma_1$ and $\sigma_2$ are Reimann sums of $f$ over partitions P$_1$ and P$_2$ of $[a,b]$ with norms less than $\delta$.

First Recalling the primary definition: As $\int_{a}^{b} f(x)$ exists, there is a unique $L$ s.t. $L = \int_{a}^{b} f(x)$ and for every $\epsilon>0$, there is a $\delta >0$ s.t. $$|\sigma - L| < \epsilon$$