Let $X,Y$ be normed space. Assume there exists $c_1 , c_2>0$ such that $$ c_1 \| x \|_X \leqslant \| x \|_Y \leqslant c_2 \| x \|_X. $$ Then if $ \| x_1 \|_Y \leqslant \| x_2 \|_Y$ then $\|x_1 \|_X \leqslant \|x_2 \|_X$ holds? I think this does not hold by using only the definition of the equivalence of norms.
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No, of course it doesn't hold. In fact your property would imply that there is some constant $c$ uch that $\|x\|_X = c \|x\|_Y$ for all $x$.