TIFR CSE 2024 | Part A | Question: 12

Question

A subset $\text{S}$ of the rational numbers is said to be "nice" if for every infinite sequence of $x_1, x_2, \ldots$ of elements from $\text{S}$, there is always two indices $i<j$ such that $x_i \leq x_j$. Consider the following statements.

1. The set of natural numbers $\mathbb{N}$ is "nice".
2. The set of integers $\mathbb{Z}$ is "nice".
3. The set of positive rational numbers is "nice".

Which of the above statements is/are true?

• A. Only $\text{(i)}$.
• B. Only $\text{(i)}$ and $\text{(ii)}$.
• C. Only $\text{(i)}$ and $\text{(iii)}$.
• D. All three statements are true.
• E. None of the three statements is true.