$\lim_{|P|\to0} \sum_{i=1}^n f(\xi_i)g(\eta_i)(t_{i}-t_{i-1})=\int_a^b f(x)g(x)dx$ Let $f,g:[a,b]\to\Bbb{R}$ integrable functions. Let's dot each partition $P$ in two different ways, choosing in all the interval $[t_{i-1},t_i]$ a point $\xi_i$ and a dot $\eta_i$. Show that:
$$\lim_{|P|\to0} \sum_{i=1}^n f(\xi_i)g(\eta_i)(t_{i}-t_{i-1})=\int_a^b f(x)g(x)dx$$
I need something as accurate as possible, because i really need to understand this question and i'm stuck at the very beginning.
 A: Let's first understand the question and to understand that we first need to write the definition of Riemann integrability.

Riemann Integrability: Let $f$ be defined and bounded on $[a, b]$. A set $P$ of the form $$P = \{x_{0}, x_{1}, x_{2}, \dots, x_{n}\}$$ is said to be a partition of $[a, b]$ if $$a = x_{0} < x_{1} < x_{2} < \dots < x_{n} = b$$ and the norm (or mesh) of $P$ (denoted by $|P|$) is defined by $|P| = \max_{i = 1}^{n}(x_{i} - x_{i-1})$. A sum of the form $$S(f, P) = \sum_{i = 1}^{n}f(\xi_{i})(x_{i} - x_{i - 1})$$ is called a Riemann sum for $f$ over partition $P$ where $\xi_{i}$ is any point in $[x_{i - 1}, x_{i}]$. The function $f$ is said to be Riemann integrable on $[a, b]$ if there is a number $I$ with the following property: For any given $\epsilon > 0$ there is a number $\delta > 0$ such that $$|S(f, P) - I| < \epsilon$$ whenever $|P| < \delta$. The number $I$ is called the Riemann integral of $f$ on $[a, b]$ and denoted by $\int_{a}^{b}f(x)\,dx$.

The condition mentioned in the above definition is stated symbolically as $$\lim_{|P| \to 0}S(f, P) = \lim_{|P| \to 0}\sum_{i = 1}^{n}f(\xi_{i})(x_{i} - x_{i - 1}) = \int_{a}^{b}f(x)\,dx$$ It is a standard result in the theory of Riemann integration that if $f, g$ are Riemann integrable over $[a, b]$ then their product $fg$ is also Riemann integrable over $[a, b]$. This means that $$\lim_{|P|\to 0}\sum_{i = 1}^{n}f(\xi_{i})g(\xi_{i})(x_{i} - x_{i - 1}) = \int_{a}^{b}f(x)g(x)\,dx$$ Your question asks to prove that in the above equation we can choose the points $\xi_{i}$ for $f$ and $g$ separately i.e. we can choose points $\xi_{i}$ for $f$ and points $\eta_{i}$ for $g$.
One can prove this by using the following criterion for Riemann integrability

Criterion for Riemann integrability: Let $f$ be defined and bounded on $[a, b]$. If $$P = \{x_{0}, x_{1}, x_{2}, \dots, x_{n}\}$$ then we define upper and lower Darboux sums $$U(f, P) = \sum_{i = 1}^{n}M_{i}(x_{i} - x_{i - 1}), L(f, P) = \sum_{i = 1}^{n}m_{i}(x_{i} - x_{i - 1})$$ where $$M_{i} = \sup\,\{f(x): x \in [x_{i - 1}, x_{i}]\},\,m_{i} = \inf\,\{f(x): x \in [x_{i - 1}, x_{i}]\}$$ The function $f$ is Riemann integrable on $[a, b]$ if and only if for every $\epsilon > 0$ there is a $\delta > 0$ such that $$U(f, P) - L(f, P) < \epsilon$$ for all partitions $P$ of $[a, b]$ with $|P| < \delta$.

Since $f$ is Riemann integrable on $[a, b]$ we have an $M$ such that $|f(x)| < M$ for all $x \in [a, b]$. Further note that $g$ is Riemann integrable and hence the above criterion applies. Therefore given any $\epsilon > 0$ there is a $\delta_{1} > 0$ such that $$U(g, P) - L(g, P) < \frac{\epsilon}{2M}$$ for all partitions $P$ with norm less than $\delta_{1}$.
Next let $I = \int_{a}^{b}f(x)g(x)\,dx$ and then we have a number $\delta_{2} > 0$ such that $$\left|\sum_{i = 1}^{n}f(\xi_{i})g(\xi_{i})(x_{i} - x_{i - 1}) - I\right| < \frac{\epsilon}{2}$$ whenever $|P| < \delta_{2}$. Now we can see that if $|P| < \delta = \min(\delta_{1}, \delta_{2})$ then we have
\begin{align}A &= \left|\sum_{i = 1}^{n}f(\xi_{i})g(\eta_{i})(x_{i} - x_{i - 1}) - I\right|\notag\\
&\leq \left|\sum_{i = 1}^{n}f(\xi_{i})g(\eta_{i})(x_{i} - x_{i - 1}) - \sum_{i = 1}^{n}f(\xi_{i})g(\xi_{i})(x_{i} - x_{i - 1})\right|\notag\\
&\,\,\,\,\,\,\,\,+ \left|\sum_{i = 1}^{n}f(\xi_{i})g(\xi_{i})(x_{i} - x_{i - 1}) - I\right|\notag\\
&< \sum_{i = 1}^{n}|f(\xi_{i})||g(\eta_{i}) - g(\xi_{i})|(x_{i} - x_{i-1}) + \frac{\epsilon}{2}\notag\\
&< M\{U(g, P) - L(g, P)\} + \frac{\epsilon}{2}\notag\\
&< M\cdot\frac{\epsilon}{2M} + \frac{\epsilon}{2}\notag\\
&= \epsilon\notag
\end{align}
and this means that $$\lim_{|P|\to 0}\sum_{i = 1}^{n}f(\xi_{i})g(\eta_{i})(x_{i} - x_{i - 1}) = \int_{a}^{b}f(x)g(x)\,dx$$
Note: The same proof works using suitable modifications if the limit is taken via refinement of partitions instead of reducing their norms to $0$.
