Let $(X,d)$ be a nonseparable metric space. It follows from here that there exists an uncountable subset $N$ of $X$ and $r>0$ such that for all $x\neq y$ in $N$ we have $d(x,y)>r$.
Is it possible to show that such $N$ must have cardinality greater than or equal to $\mathbb R$?
Thanks a lot for your help.
It depends upon whether or not you accept the continuum hypothesis. If you do, then the statement is automatically true, because then the cardinal of any uncountable is greater than or equal to the cardinal of $\Bbb R$.
Otherwise, take an uncountable set $X$ whose cardinal is smaller than that of $\Bbb R$, consider on it the discrete metric and take $N=X$. Then, if $x,y\in N$ and $x\ne y$, $d(x,y)=1>\frac12$.