Solutions

The sequence

$$a_{n} = (1,0,1,0,1,\dots)$$

is a clear example of a vercongent sequence. If we set $x = \frac{1}{2}$ and choose $\epsilon > 1$, then

$$\begin{aligned} \forall N \in \mathbb{N}, n \geq N &\implies \abs{ x_{n} - \frac{1}{2} } < \epsilon \\ &\implies \abs{1 - \frac{1}{2}} < \epsilon \; \wedge \; \abs{0 - \frac{1}{2}} < \epsilon \\ &\implies \frac{1}{2} < 1 < \epsilon \end{aligned}$$