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tags: math, calculus, real analysis
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Question:
Suppose functions $f(x)$ and $g(x)$ continuous on point $p\in I \subset \mathbb{R}$ where $I$ is some closed interval.
Given any sequence of intervals $\{I_i\}$ with $\lim\limits_{i \to \infty}{\lvert I_i \rvert}=0$ and $\forall i \left(p \in I_i \subset I \right)$, consider the value $\int\limits_{I_i}{ \left( f(x)g(x) \right) }dx$ and $g(p)\int\limits_{I_i}{\left( f(x) \right)}dx$
Prove or disprove the following proposition:
$\lim\limits_{i \to \infty}
\dfrac{
\left( \int\limits_{I_i}{ \left( f(x)g(x) \right) }dx \right)}{
\left( g(p)\int\limits_{I_i}{\left( f(x) \right)}dx
\right)} = 1$

Answer:
Direct result of fundamental theorem of calculus (?)