Why is: $\lim_{n\to\infty}\frac1n\sum_{r=0}^{n}\left(\frac{r}{n}\right)=\int_0^1x\,dx$ Could somebody please explain how: $$\lim_{n\to\infty}\frac1n\sum_{r=0}^{n}\left(\frac{r}{n}\right)=\int_0^1x\,dx$$
I am just beginning integration and this has been highlighted as a very fundamental and important result/theorem in my senior's notes. Does it have any name ? Any books you could suggest would also be very helpful.
Firstly, notice that $x \mapsto x$ is a continuous function on $\mathbb{R}$, and hence uniformly continuous on the closed interval $[0,1]$, thus its Riemann integral on $[0,1]$ exists, and hence can be expressed as a limit of the left/right Riemann sum using partitions with strips of width $\frac{1}{n}$ for integer $n$. Secondly, the expression inside your limit on the left-hand side is just zero plus the right Riemann sum.
The $\text{LHS}$ is area under $y=x$ from $x=0$ to $x=1$.

It divides your triangle in infinitesimal thick rectangles with width $\frac1n$ and the term $\frac r n$ divides $x=0\to1$ as height of each rectangle.

$\text{RHS}$ is also area according to fundamental theorem of calculus.