Definition of a point $x$ in a Riemann sum. $$\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta(x)$$
I am interested in what $x_k$ is. On stackexchange I have seen $x_k$ being defined as:
$$x_k = \Delta(x)(k) + a$$
Where $a$ is the initial beginning etc...
Is there are a proof of this, or is it a definition?
 A: Very briefly, $x_k$ denotes a point in the $k$th part of the partition of $[a,b]$. It doesn't actually matter which point: the limit is supposed to exist regardless of what point is chosen within the $k$th part of the partition.
Equivalently, $x_k$ denotes the left endpoint of one of the $k$th part of the partition, where the limit is over partitions whose widths go to $0$.
Often in first calculus classes, one evaluates integrals by assuming that the left endpoints of the partitions of the interval $[a,b]$ into $k$ same-sized parts is sufficient (and it is sufficient for continuous functions, for instance). In this case, the left endpoint is $a$. The next endpoint would be $a + \frac{b-a}{k}$, and so on. Is this a necessary definition? No. But it's more concrete, and sometimes having something concrete is nice in a first calculus class.
You might also be interested in a post I wrote for my students about calculus and the intuition behind some of these ideas. There's also a gif of integration by rectangles, which I rather like.
A: More generally a Riemann sum can use any partition of the interval $[a,b]$; i.e., a division of $[a,b]$ into $n$ subintervals not necessarily of the same size.
If the Riemann integral exists, it is equal to your limit on the RHS using a partition of subintervals of the same size, where $$ \Delta x = \frac{b-a}{n} \ \ \hbox{ and } \ \ x_k = a + k\Delta x$$
