Composition of Sequentially Computable Real Functions is Sequentially Computable

Theorem

Let $f,g : \R \to \R$ be sequentially computable real functions.

Let $h : \R \to \R$ be defined as:

$\map h x = \map f {\map g x}$

Then $h$ is sequentially computable.

Proof

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

As $g$ is sequentially computable:

$\sequence {\map g {x_n}}$

is computable.

As $f$ is sequentially computable:

$\sequence {\map f {\map g {x_n}}}$

is computable.

But:

$\map f {\map g {x_n}} = \map h {x_n}$

Therefore:

$\sequence {\map h {x_n}}$

is computable.

As $\sequence {x_n}$ was arbitrary, $h$ is sequentially computable by definition.

$\blacksquare$