I saw in Wikipedia that a real number $a$ is computable if there exists a computable sequence of rational numbers $q_n$ converging to $a$ such that $|q_n - q_{n+1}| < 2^{-n}$, for each $n$.
I just want to know if this is correct, because I couldn't find anywhere else such definition.
Yes, it is correct. Not sure what is the reference you are looking at, but I would say that this definition is rather widespread. Wikipedia cites Weihrauch's book [1] in the end. I may add Pour-El and Richard's book [2,definition 3].
[1]: Weihrauch, Klaus, Computable analysis. An introduction, Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer. x, 281 p. (2000). ZBL0956.68056.
[2]: Pour-El, Marian B.; Richards, J. Ian, Computability in analysis and physics, Perspectives in Mathematical Logic, Berlin etc.: Springer-Verlag. x, 206 p. DM 128.00 (1989). ZBL0678.03027.