# What fraction of fractions does Cantor’s famous sequence enumerate?

Cantor’s famous sequence

$$11,12,21,13,31,14,23,32,41,15,51,16,...\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$$

provides a bijection between natural numbers and positive rational numbers or cancelled fractions.

About half of the fractions $$qiq_i$$ lie within $$0. What is the limit of the ratio

$$limk→∞|{x∈R|n
for $$n \in \mathbb{N}n \in \mathbb{N}$$?

Is there an $$nn$$ for which the limit is $$00$$? And if so, what is the first such $$nn$$?