# Inductive Definition of Sequence

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## Theorem

Let $X$ be a set.

Let $h \in \N$.

Let $a_i \in X$ for all $i \in \set {1, 2, \ldots, h}$.

Let $S$ be the set of all finite sequences whose codomains are in $X$.

Let $G: S \to X$ be a mapping.

Then there is a unique sequence $f$ whose codomain is in $X$ such that:

- $f_i = \begin{cases}

a_i & : i \in \set {1, 2, \ldots, h} \\ \map G {f_1, f_2, \ldots, f_{i - 1} } & : i \ge h + 1 \end{cases}$

## Proof

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## Also known as

Such a definition for a sequence is also known as a **recursive definition**.

## Sources

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Theorem $18.4$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous): Appendix $\text{A}.12$: Induction