Let $(x_n)_{n\geq 1} \subset \left[-\frac{1}{2}, \frac{1}{2} \right]$ such that $|x_m+x_n|\geq \frac{m}{n}, \forall m,n \in \mathbb{N}_{\geq1 }$ with $m<n$. Prove that $(x_n)$ converges.
We have $1=\frac{1}{2}+\frac{1}{2} \geq |x_{n}|+|x_{n-1}| \geq |x_n+x_{n-1}|\geq \frac{n-1}{n} \to 1$, so $$\lim_{n\to \infty} |x_n+x_{n-1}|=1$$ So we should prove that $x_n \to \frac{1}{2} \text{ or } -\frac{1}{2}$, but I didn't succeed in proving it.