Completeness with equivalent norm

Question

Suppose I have a subset $C$ of a Banach space $X$ which is complete respect to a given (norm induced by a) metric $d$. Let also suppose that it exist a subset $S\subset C$ which is a metric space respect to a metric $d'$ and that it holds: $$A d(x,y)\le d'(x,y)\le B d(x,y) \ \ \forall x,y\in S$$ for some positive constants $A,B\in\mathbb{R}$. Can I conclude that $(S,d')$ is also a complete metric space? Or do I need for example that $S$ is closed respect to one between $d$ and $d'$?