Question

The enumeartion of rational numbers in [0,1] has a convergent sub sequence

Answer 1

Answer:

Step-by-step explanation:

The argument given by Elio Joseph can be refined in order to prove that for any given real number, there always exists a subsequence which converges to it.

For that, let x∈R be a real number. Consider the interval [x−1,x+1]. It contains infinitely many rationals, but in particular it contains one rational, which is some an1 of our enumeration.

Now, having chosen ani, we consider the interval [x−1i+1,x+1i+1]. This contains infinitely many rationals. In particular, we have that it contains one rational which is not one of the first ni of our enumeration. We take such rational to be our ani+1.

It is clear that ani→x.