There exists a computable function which, given any positive rational error bound ε {\displaystyle \varepsilon } , produces a rational number r such that | r − a | ≤ ε . {\displaystyle |r-a|\leq \varepsilon .} There is a computable sequence of rational numbers q i {\displaystyle q_{i}} converging to a {\displaystyle a} such that | q i − q i + 1 | < 2 − i {\displaystyle |q_{i}-q_{i+1}|<2^{-i}\,} for each i.