Sum of Computable Real Sequences is Computable

Theorem
Let $\sequence {x_n}$ and $\sequence {y_n}$ be computable real sequences.

Then $\sequence {x_n + y_n}$ is a computable real sequence.

Proof
By definition, there exist total recursive $f,g : \N^2 \to \N$ such that:
 * For every $m,n \in \N$, $\map f {m, n}$ and $\map g {m, n}$ respectively code integers $k$ and $\ell$ such that:
 * $\dfrac {k - 1} {n + 1} < x_m < \dfrac {k + 1} {n + 1}$
 * $\dfrac {\ell - 1} {n + 1} < y_m < \dfrac {\ell + 1} {n + 1}$

Define $h : \N^2 \to \N$ as:
 * $\map h {m, n} = \map {\operatorname{quot}_\Z} {\map f {m, 4 n + 3} +_\Z \map g {m, 4 n + 3} +_\Z 2, 4_\Z}$

which is total recursive by:
 * Addition is Primitive Recursive
 * Multiplication is Primitive Recursive
 * Constant Function is Primitive Recursive
 * Addition of Integers is Primitive Recursive
 * Multiplication of Integers is Primitive Recursive
 * Quotient of Integers is Primitive Recursive
 * Primitive Recursive Function is Total Recursive Function

Now, let $m, n \in \N$ be arbitrary.

Let $k'$ and $\ell'$ be the integers coded by $\map f {m, 4 n + 3}$ and $\map g {m, 4 n + 3}$, respectively.

Then, $\map h {m, n} = \floor {\dfrac {k' + \ell' + 2} 4}$.

Thus, by Properties of Floor Function:
 * $\dfrac {k' + \ell' - 2} 4 < \map h {m, n} \le \dfrac {k' + \ell' + 2} 4$

Hence:
 * $\map h {m, n} - 1 \le \dfrac {k' + \ell' - 2} 4$
 * $\dfrac {k' + \ell' + 2} 4 < \map h {m, n} + 1$

From the inequalities above, we have:

Thus, $\sequence {x_n + y_n}$ is computable by definition.