Why must a normed space $X$ homeomorphic to a complete metric space $Y$ be complete?
I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am not sure how to proceed in general. Can we use an adapted open-balls argument even when Y isn't normed?
Maybe there's an equivalent condition for completeness that will come in useful? In a normed space we can use (absolutely convergent)$\implies$(convergent) for series, but this won't work in a metric space.
I've tried using an explicit homeomorphism $f:X\rightarrow Y$ and taking pullbacks and pushforwards along $f$ of a Cauchy sequence in $X$, but without success.
Many thanks for any help with this!