Composition of Sequence with Mapping

Theorem
Let $$\left \langle {a_j} \right \rangle_{j \in B}$$ be a sequence.

Let $$\sigma: A \to B$$ be a mapping, where $$A \subseteq \mathbb{N}$$.

Then $$\left \langle {a_j} \right \rangle \circ \sigma$$ is a sequence whose value at each $$k \in A$$ is $$a_{\sigma \left({k}\right)}$$.

Thus $$\left \langle {a_j} \right \rangle \circ \sigma$$ is denoted $$\left \langle {a_{\sigma \left({k}\right)}} \right \rangle_{k \in A}$$.