Let $\displaystyle (R,d)$ be a metric space. Is it true, that for any metric $\displaystyle d$, from $\displaystyle d(x_{n},x)\longrightarrow 0$ follows $\displaystyle d(x_{n}-x,0)$ converges, when $\displaystyle n\longrightarrow\infty$?

Actually, it is not true, but it is terribly difficult to find counterexample