What are convergent sequences in a given metric space?

Question

Let on $ \mathbb{R} $ be given a metric $d$, $$d(x, y) =\begin{cases} |x-y| +1& x > 0 \text{ or(exclusive or) } y>0 \\ |x-y|& \text{otherwise} \end{cases} $$ What are convergent sequences in that space? I think that the only problem is with sequences that converge towards 0. They do not converge here? Because $\epsilon - 1 $ does not have to be positive?

Answer 1

A sequence will converge to $0$ iff it converges to $0$ from the left, i.e. if it converges to $0$ in the standard metric and contains finitely many positive members.