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

Cantor’s famous sequence

$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$

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

About half of the fractions $q_i$ lie within $0 < x \leq 1$. What is the limit of the ratio

$\lim\limits_{k\to\infty}\frac{|\{x ∈ \mathbb{R}| n < x \leq n+1\} ∩ \{q_1, q_2, ..., q_k\}|}{|\{x ∈ \mathbb{R} | 0 < x \leq 1\} ∩ \{q_1, q_2, ..., q_k\}|}$
for $n \in \mathbb{N}$?

Is there an $n$ for which the limit is $0$? And if so, what is the first such $n$?

Answer

Attribution
Source : Link , Question Author : Hilbert7 , Answer Author : Community