Cantor’s famous sequence

11,12,21,13,31,14,23,32,41,15,51,16,...

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

About half of the fractions qi lie within 0<x≤1. What is the limit of the ratio

limk→∞|{x∈R|n<x≤n+1}∩{q1,q2,...,qk}||{x∈R|0<x≤1}∩{q1,q2,...,qk}|

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*