Let $a,b > 0$, and consider the interval $(a,b)$. Let $a < x_0 < b$. Prove that (a,b) is a neighborhood of $x_0$.
Proof: Assume $a,b>0$ and $a < x_0 < b$. Let $\epsilon = \frac{b-a}{2}$. Then consider the interval $(x_0-\frac{b-a}{2}, x_0 + \frac{b-a}{2})$. Then $(x_0-\frac{b-a}{2}, x_0 + \frac{b-a}{2}) \subset (a,b)$. Thus $(a,b)$ is a neighborhood of $x_0$.
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