Should I explicitly declare the range after such that in a set?

Question

$S = \{(a_k)_{k = 0}^{2} \mid a_{i + 1} - a_i = 1\}$ is a set of infinite elements, and each element is consist of a 3-tuple in which an element is increased by $1$ than the previous one. However, should I explicitly declare the range after such that in the set to be $S = \{(a_k)_{k = 0}^{2} \mid a_{i + 1} - a_i = 1 , \forall 0 \leqslant i < 2\}$? In my opinion, the previous $S$ is clear enough and does not produce any ambiguity. Is there any regulation about this?