Prove that $\int_{a}^{b} h(x)dx = \lim_{k\to\infty} \sum_{j=1}^{k} h(x^{k}_{j})\frac{b-a}{k}$ using given intermediate points.

Question

CONDITIONS

• $h : [a,b] \rightarrow \mathbb{R}$ is a continuous function.

• For each $k\in\mathbb{N}$, let $x^{k}_{1},...,x^{k}_{k}\space$ be points in $\mathbb{R}$ such that for each $j\leq k$ we have

$$a+\frac{(j-1)(b-a)}{k} \leq x^{k}_{j} < a + \frac{j(b-a)}{k}.$$ Note that the $k$ represents an index and not a power.

Indeed the proof itself can be done with the Fundamental Theory of Calculus Proof but that's with end points given. How can this be proven through the points given above?