Question

Let a, b ∈ R with a < b. Let Ta,b ⊆ Q be the subset defined by Ta,b := n r ∈ Q : a < r < bo . Prove that the set Ta,b is infinite.

Answer 1

Let t $\in$ Ta,b, that is, $t_1\in$ Q with $a \lt t_1\lt b.$Then, consider the denseness of Q in R with $a \lt t_1,$ there exist $\lt t_2 \in Q$ such that $a \lt t_2 \lt t_1.$ Subsequently, there exist infinite sequences $t_n \in Q$ with $a \lt... \lt t_{n+1} \lt t_n \lt t_{n+1} \lt ... \lt t_2 \lt t_1.\\$Such that $t_n \in Ta, b$ showing that Ta, b is infinite