Let $(x_n)$ be a sequence of real numbers. Suppose that there is a real number $x_0\in\mathbb{R}$ such that for any $\varepsilon>0$ there is an index $n_\varepsilon$ such that $|x_0-x_n|<\varepsilon^2$ for all $n>n_\varepsilon$.
How does this imply that $(x_n)$ converges to $x_0$? I.e. does this imply that there is an index number $N_\varepsilon$ such that $|x_0-x_n|$ for all $n>N_\varepsilon$?