Prove that for each $n \in \mathbb{N},$ the set $P = \{j/n\}$ is a partition of $[0,1]$

Question

$$P=\left\{\frac{j}{n}:j=0,1,...n\right\}$$ I can't find anywhere in my book where it is proven that a given set is a partition. It just jumps to showing, using partitions, that a function is integrable over the interval.

Answer 1

Any finite set of points $$ a = x_1 < x_2 < \cdots < x_n = b $$ defines a partition of the interval $[a,b]$. In some contexts when discussing integration the partition might be described by the set of subintervals $[x_i, x_{i+1}]$ but the essential meaning is the same.

Those subintervals don't strictly partition the whole interval, since they are not disjoint (they share endpoints) but that does not matter when using partitions to define integrals.