Term of Computable Real Sequence is Computable

Theorem
Let $\sequence {x_i}$ be a computable real sequence.

Then, for each $i \in \N$:
 * $x_i$ is a computable real number.

Proof
By definition of computable real sequence, there exists a total recursive function $f : \N^2 \to \N$ such that:
 * For every $m,n \in \N$, $\map f {m, n}$ codes an integer $k$ such that:
 * $\dfrac {k - 1} {n + 1} < x_m < \dfrac {k + 1} {n + 1}$

Let $g : \N \to \N$ be defined as:
 * $\map g n = \map f {i, n}$

By: it follows that $g$ is a total recursive function.
 * Constant Function is Primitive Recursive
 * Primitive Recursive Function is Total Recursive Function

Additionally, for every $n \in \N$, $\map g n$ codes an integer $k$ such that:
 * $\dfrac {k - 1} {n + 1} < x_i < \dfrac {k + 1} {n + 1}$

Thus, by definition, $x_i$ is a computable real number.