Uniqueness of a Point in Neighborhoods

Theorem

Let $a\in \mathbb{R}$.

Then $x\in V_{\epsilon }(a)⟹x=a,\forall \epsilon >0$

Remark

We can make sense of this theorem by imagining even if $\epsilon$ (or the distance between $x$ and $a$) is infinitesimally small, but $x$ is still in $V_{\epsilon }(a)$.

This can only happen if the distance between $x$ and $a$ is $0$, which means $x=a$.