# Is it true that for any sequence of irrational numbers <mo fence="false" stretchy="false">{

Is it true that for any sequence of irrational numbers $\left\{{h}_{n}\right\}$ converges to zero in $\left[-h,h\right]\phantom{\rule{thinmathspace}{0ex}}$ both $\phantom{\rule{thinmathspace}{0ex}}\alpha +{h}_{n}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\alpha -{h}_{n}\phantom{\rule{thinmathspace}{0ex}}$ and $\phantom{\rule{thinmathspace}{0ex}}\alpha +{h}_{n}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\alpha -{h}_{n}\phantom{\rule{thinmathspace}{0ex}}$ are always irrationals ?
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SweallySnicles3
Most certainly not.
In fact, take any sequence ${q}_{n}$ of rational numbers in $\left[\alpha -h,\alpha +h\right]$ that converges to $\alpha$ $\left[1\right]$. Now, define ${h}_{n}={q}_{n}-\alpha$. Then,
${h}_{n}$ is in $\left[-h,h\right]$.
${h}_{n}$ converges to $0$.
for all $n\in \mathbb{N}$, hn is irrational
However, $\alpha +{h}_{n}={q}_{n}$, which means that $\alpha +{h}_{n}$ is always rational.