Let $(M,\tau)$ be a metric space and $(Y,\tau_{Y})$ a subspace of a the metric space with an equivalent metric.
Prove: for all $\{x_{n}\}\subseteq Y$ and $x\in Y$, $x_{n}\rightarrow x$ according to $\tau\iff x_{n}\rightarrow x$ according to $\tau_{Y}$
The given proof is: if $x_{n},x\in Y$ so $\tau(x_{n},x)=\tau_{Y}(x_{n},x)$ and by the definition of equivalent of metrics $\tau(x_{n},x)\rightarrow 0\iff\tau_{Y}(x_{n},x)\rightarrow 0$
What does it follow from the "definition of equivalent of metrics"?
because in the case that all of the elements are form the subspace of the metric space both $\tau$ and $\tau_{Y}$ "behave" the same?
what if we only knew that $\tau(x_{n},x)\rightarrow 0$ when $x_{n},x\in M$ and $x_{n},x\notin Y$, can we still say that $\tau_{Y}(x_{n},x)\rightarrow 0$?