Constant Sequence of Rational Number is Computable
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Theorem
Let $r \in \Q$ be a rational number.
Let $\sequence {x_n}$ be defined as:
- $x_n = r$
Then, $\sequence {x_n}$ is a computable rational sequence.
Proof
By Existence of Canonical Form of Rational Number, let:
- $r = \dfrac p q$
where:
- $p \in \Z$
- $q \in \Z_{>0}$
Let $m$ be the code number for the integer $p$.
As $q \ge 1$, it follows that:
- $q - 1 \in \N$
We define $N, D : \N \to \N$ as:
- $\map N n = m$
- $\map D n = q - 1$
which are total recursive by:
Then:
- $\dfrac {k_n} {\map D n + 1} = \dfrac p q = r = x_n$
where $\map N n = m$ codes the integer $k_n = p$
Therefore, $\sequence {x_n}$ is computable by definition.
$\blacksquare$