I've been assigned the following problem for my Real Analysis class and I'm not sure how to go about it.
"Suppose that $\int_{a}^{b}f(x)dx$ exists and $\exists$ A such that, $\forall \epsilon>0$ and $\delta>0$, there is a partition P of [a,b] with $\left\Vert P \right\Vert < \delta$ and a Riemann sum $\sigma$ of $f$ over P that satisfies the inequality $\left|\sigma -A\right|<\epsilon.$ "
Show that $\int_{a}^{b}f(x) dx = A$
EDIT:
My work:
Since $f$ is defined on $[a,b]$ and $\exists A$ such that $\forall \epsilon>0, \exists \delta >0$, then $\left|\sigma -A\right|<\epsilon$.
Also, since $\sigma$ is any Riemann sum of $f$ over partition P of [a,b] such that $\left\Vert P \right\Vert < \delta$
I can say that $A$ is the Riemann Integral of $f$ over [a,b],
or that, $$\int_{a}^{b}f(x) dx = A$$