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 $d'$?
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Hint. Show that the Cauchy sequences for one metric are the same as the Cauchy sequences for the other metric. – GEdgar Apr 14 '18 at 12:22
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This is true, but the limit which is assured respect to d could not stay in S. Isn't it? So I have to ask S being closed. – aleio1 Apr 14 '18 at 13:01