Identity Function is Sequentially Computable Real Function

Theorem

Let $I_\R : \R \to \R$ denote the identity function on $\R$.

Then $I_\R$ is a sequentially computable real function.

Proof

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

By definition of identity function:

$\map {I_\R} x = x$

Therefore:

$\sequence {\map {I_\R} {x_n}} = \sequence {x_n}$

and is thus computable.

As $\sequence {x_n}$ was arbitrary, it follows that $I_\R$ is sequentially computable by definition.

$\blacksquare$