(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find all accumulation points of the following sequence:

an enumeration of all rational numbers in (0,1)

2. Relevant equations

Boltzano-Weirstrass Theorem:

Every bounded sequence has a convergent sequence (hence, an accumulation point)

3. The attempt at a solution

Because the enumeration of all rational numbers in (0,1) is bounded, it must have at least one convergent sequence. But if there is an accumulation point for the rational numbers in (0,1) there must also be an accumulation point for the rational numbers in (0,0.5), and the logic continues so there must be infinitely many accumulation points in (0,1). So are the accumulation points every rational number in (0,1)?

This seems odd to me that any rational number can be an accumulation point

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Accumulation point of rational numbers in (0,1)

**Physics Forums | Science Articles, Homework Help, Discussion**