Strictly_Increasing_Sequence_induces_Partition

Let $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le n}$ be a strictly increasing finite sequence of natural numbers. Let $\forall k \in \left[{1 \,.\,.\, n}\right]: A_k := \left[{r_{k-1} + 1 \,.\,.\, r_k}\right]$ Then $\left\{{A_k: k \in \left[{1 \,.\,.\, n}\right]}\right\}$ is a partition of $\left[{r_0 + 1 \,.\,.\, r_n}\right]$.