Some real number $x \in\mathbb R$ is computable if there is a computable function $f :\mathbb N \to\mathbb Q$ such that, for all $r \in\mathbb N,$
$$|f(r) − x| ≤ 2^{−r}$$
If $x, y \in R$ and $x$ and $y$ are computable, then prove $x+y$ is computable.
I understand we are supposed to prove $|f(r) − (x+y)| ≤ 2^{−r}$ by defining a function $f$ and then use the fact that $x$ and $y$ are computable to prove it, but I don't get how to start the problem.
Compute $x$ and $y$ to $2^{-(r+1)}$ absolute precision. Then $x+y$ has error at most $2^{-r}$.