How to prove the following property for this set (see details below, please)?

Question

Let A be an uncountable subset of R. Then, A contains a subset S, composed of at least 2 elements, with the so called "in between" property, that is: for every distinct elements x and y of S, there exists z in S such that x < z < y.

Answer 1

Normally I've heard the word "connected" in place of "in-between property", but I'll go with the phrasing in the question.

Proceed by contradiction. Suppose that there exists no subset S of A such that S has at least 2 elements and S has the "in-between" property. Then A contains no intervals of the real line, since an interval of the real line has the "in-between" property. So A is a set of disconnected points.

But a set of disconnected points is countable. (For example, you could count them in order from least absolute value to greatest absolute value). So A is countable. This contradicts the hypothesis that A is uncountable. Therefore A must contain a subset S with at least 2 elements and the "in-between" property.

[In fact you could prove that the subset S is uncountable itself.]