Composition of Sequence with Mapping

Theorem
Let $\sequence {a_j}_{j \mathop \in B}$ be a sequence.

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

Then $\sequence {a_j} \circ \sigma$ is a sequence whose value at each $k \in A$ is $a_{\map \sigma k}$.

Thus $\sequence {a_j} \circ \sigma$ is denoted $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$.

Proof
By definition, a sequence is a mapping whose domain is a subset of $\N$.

Let the range of $\sequence {a_j}_{j \mathop \in B}$ be $S$.

Thus $\sequence {a_j}_{j \mathop \in B}$ can be expressed using the mapping $f: B \to S$ as:
 * $\forall j \in B: \map f j = a_j$

Let $k \in A$.

Then $\map \sigma k \in B$.

By definition of composition of mappings: