Solutions

Suppose $a_{n} \rightarrow a$ and $a_{n} \rightarrow b$ with $a \neq b$. Therefore, if we choose $\epsilon = |a - b| > 0$, then

$$\exists N \in \mathbb{N}, \forall n \in \mathbb{N}, n \geq N \implies a_{n} \in V_{|a - b|}(a).$$

This would imply that the sequence $a_{n}$ does not converge to $b$ since $b$ does not belong to $V_{\abs{a - b}}(a)$ and we thus have a contradiction.