Converging sequence of irrationals

Question

Let $x_{0}\in\mathbb{R}$ and $\eta>0$. Find a sequence $\left(x_{n}\right)_{n=1}^{\infty}$ that satisfies the following:

1. $\forall n\in\mathbb{N} \quad x_{n}\in (x_{0}-\eta,x_{0}+\eta)\setminus\{x_{0}\}$
2. $\forall n\in\mathbb{N} \quad x_{n}\in \mathbb{R}\setminus\mathbb{Q}$
3. $\underset{n\rightarrow\infty}{\lim}x_{n}=x_{0}$

Any idea on how to even begin to create such a sequence? I've thought about creating something recursively, but I couldn't find something that assures that I won't get a rational member of the sequence.

Answer 1

Hint:

Create a rational sequence that converges to $x_0$, and add an irrational vanishing term like

$$\frac{1}{\sqrt2\,n\left\lceil\dfrac1\eta\right\rceil}.$$