Let $\displaystyle \frac{x}{2}=\varepsilon>0$. By $\displaystyle x_n\to x$ there exists some $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\leqslant n\implies |x_n-x|<\varepsilon\implies x-\varepsilon=\frac{x}{2}<x_n$