Reciprocal of Computable Real Sequence is Computable

Theorem

Let $\sequence {x_m}$ be a computable real sequence.

Suppose that, for all $m \in \N$:

$x_m \ne 0$

Then:

$\sequence {\dfrac 1 {x_m}}$

is a computable real sequence.

Proof

Lemma

There exists a total recursive function $\psi : \N \to \N$ such that, for all $m, p \in \N$ and $\alpha \in \R$, if:

$p \ge \map \psi m$

and:

$\size {x_m - \alpha} \le \dfrac 1 {p + 1}$

it follows that:

$\size \alpha > \dfrac {\size {x_m}} 2$

$\Box$

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