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


This needs considerable tedious hard slog to complete it.
In particular: tedious
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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
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