TITLE: Equality of limits with respect to different metrics.
QUESTION [5 upvotes]: Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in  X$ with respect to metric $d_1$. Does it follow that $\{x_n\}_{n\in\mathbb{N}}$ either converges to $x$ with respect to $d_2$ or does not converge at all? If the answer is no, can we have some additional requirements on $d_1$ and $d_2$ for it to be the case?

REPLY [4 votes]: Here are example which show that even when $X$ is a normed space and the distances come from a norm, the result may not hold.