Equal in Distributions

$\frac{1}{\sqrt{3}}$ is an irrational number, so we can apply a consequence of Weyl's criterion . The sequence $\left \{ \dfrac k{ \sqrt{3}} \right \}$ is equidistributed,and hence dense in $[0,1]$ . This means, for example that, althought $0.75$ and $1 - 0.75 = 0. 25$ don't belong to the sequence $\left \{ \dfrac k{\sqrt{3}} \right \}$ , in each neighborhood (ball of radius r > 0) of $0.75$ contained in $[0,1]$ there exist and there are "the same numbers" of terms of this sequence as in in a neighborhood (ball of the same radius r > 0) of $0.25$ contained in $[0,1]$ . Hence for $0.75 + 0.25 = 1$ we are using 2 terms of this sequence, so we can say the limit of aritmethic mean of this sequence (equal to the limit above) is $1/2$ .