Consider $f:[a,b]\to \mathbb{R}$ is a continuous and bounded function. Assume $f(x)\geq 0,\enspace \forall x\in [a,b].$Divide $[a,b]$ into $n$ equal intervals. Define the lower sum $L_n$ and upper sum $U_n$ approximations as per the given n equal sub-intervals. Show that $\lim_{n\to \infty}U_n-L_n=0$ and $\lim_{n\to \infty}L_n=\lim_{n\to \infty}U_n=A_R.$$A_R$ is the area under $f$ on $[a,b].$
Following is my approach:
Let $m_i$ and $M_i$ be the minimum and maximum value of $f(x)$ in the interval $[x_{i-1},x_i]$.
Consider a partition P with $x_0=a$ and $x_n=b$, $x_i=a+(\frac{b-a}{n})i$
$$L_n=\sum_{i=1}^{n}m_i(x_i-x_{i-1})$$
$$U_n=\sum_{i=1}^{n}M_i(x_i-x_{i-1})$$
$$U_n-L_n=\sum_{i=1}^{n}(M_i-m_i)(x_i-x_{i-1})$$
$$\implies U_n-L_n=\frac{b-a}{n}\sum_{i=1}^{n}(M_i-m_i)$$
We know that for $i^{th}$ sub-interval $M_i\geq m_i$ as we make more and more intervals, the values $M_i$ and $m_i$ approach each other; and finally on applying limit $n\to \infty$ $\enspace M_i$ and $m_i$ become equal.
$\implies \lim_{n\to \infty}U_n-L_n=\frac{b-a}{n}\sum_{i=1}^{n}(M_i-m_i)=0$.
Now the above proof looks reasonable, but the problem is it is not mathematically rigorous.Especially the last part of the proof just uses statements of which seems obvious. But what I want is a rigorous approach of the last part of the proof.I tried a lot but could not succeed. Any help would be appreciated.
Also I am unable to present the proof for the second part of the question.