Sequences of Irrational numbers

Question

Let $ \alpha $ is fixed irrational number. Let $ [-h,h] $ be an interval. Is it true that for any sequence of irrational numbers $ \{h_n\} $ converges to zero in $ [-h,h] \, $ both $ \, \alpha + h_n \, \mbox{and} \, \alpha - h_n \, $ are always irrationals ?

Answer 1

Most certainly not.

In fact, take any sequence $q_n$ of rational numbers in $[\alpha - h, \alpha + h]$ that converges to $\alpha$ [1]. Now, define $h_n=q_n-\alpha$. Then,

1. $h_n$ is in $[-h,h]$.
2. $h_n$ converges to $0$.
3. for all $n\in\mathbb N$, $h_n$ is irrational

However, $\alpha + h_n = q_n$, which means that $\alpha+h_n$ is always rational.

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[1] There are plenty such sequences, for example, $q_n=\alpha + \frac{1}{n+M}$ where $M$ is large enough so that $q_1\in[\alpha-h,\alpha+h]$).