$ a_1, a_2, ...... ,a_n $ are real numbers. Prove that there is a real number $ k$ such that $ a_1 + k, a_2+k,\ldots, a_n +k $ all are irrational.

Question

$ a_1, a_2, ...... ,a_n $ are real numbers. Prove that there is a real number $ k$ such that $ a_1 + k, a_2+k,......, a_n +k $ all are irrational.

My thinking: If $ a_1, a_2, ...... ,a_n $ these are real numbers and $ k$ is irrational then the statement holds good. And If $ k$ and $ a_1, a_2, ...... ,a_n $ are irrational then also it's hold good. But if both are not irrational nor transcendental number then how can we get $ a_1 + k, a_2+k,......, a_n +k $ this as irrational number? Please help me. Thanks in advance.

Answer 1

This follows from a simple counting argument.

For a fixed $i$ there are only countably many $k$ such that $a_i+k$ is rational; call this set $S_i$. But then $S= \bigcup S_i$ is also countable, so it can not be all of $\mathbb R$. Any $k\in \mathbb R-S$ will work.

Remark: the same argument works even if you have a countable collection $\{a_i\}$.