Dual set of the unit ball is part of the unit ball. Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$.
Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$.
I'm attempting to prove that for any set $C\subseteq\mathbb{R}^d$ it holds that $C=B$ if and only if $C=C^*$. I've managed to prove the implication from the right to the left by the Cauchy-Schwarz inequality.
However, I'm having some trouble with the converse implication. In particular, how to prove that $C=B$ implies $C^*\subseteq C$?

Hint:
For $y \in B^*$, notice that if $y \neq 0$ then $\frac1{\|y\|} y \in B$.
