Definition. Function $ d:X\times X\rightarrow \mathbb{R} $ is called metric, if d satisfes following axioms:
- $\quad d(x,y)=0\Longleftrightarrow x=y$
- $\quad d(x,y)=d(y,x)$
- $d(x,y)\leq d(x,z)+d(z,y)$
$\forall x,y,z\in X$. Metric space is denoted as $(X,d)$
Let $ (\mathbb{R},d) $ be metric space. Is it true, that for any metric $ d $, from $ d(x_{n},x)\longrightarrow0 $ follows $ d(x_{n}-x,0)\longrightarrow C $, when $ n\longrightarrow\infty $, where $C\in \mathbb{R}$ is constant? Actually, it is not true, but it is terribly difficult to find counterexample