Q:

If $x$ is a limit point of a non empty subset $A$ of a metric space $X$, must it also be a limit point of $Y$ where $A\subset Y\subset X$

If $x$ is a limit point of a non empty subset $A$ of a metric space $X$, must it also be a limit point of $Y$ where $A\subset Y\subset X$?
This seems to be trivially true to me. If $x$ is a limit point of $A$ then for any $r > 0$ there exists $B_r(x)$ such that this open ball will contain at least one point of $A$ other than $x$ itself.
As $A \subset Y$ this open ball will clearly contain at least one point in $Y$ other than $x$ itself as all points in $A$ are in $Y$.
Is that correct or am I missing something?

A:

Yes, you are right.
Another (maybe more natural) way of thinking to limit points in metric spaces is that 

$x$ is a limit point of $A$ iff $\exists a_n\in A\setminus\{x\}$: $\lim a_n = x $

