I am trying to prove the following statement
If $C_1 \subset C_2$ then $C_2^{*} \subset C_1^{*}$
Here $C_1$ and $C_2$ are convex cones in $\mathbb{R^m}$.
The dual of a convex cone is defined as $C^{*} = \{ y : xy \leq 0 \text{ for all } x \in C \}$.
Is this a correct proof?
Suppose $y \notin C_1^{*}$, then $xy > 0$ for all $x \in C_1$. On the other hand if $x \in C_1$ then $x \in C_2$. Hence we have $y \notin C_2^{*}$. By contraposition, if $y \in C_2^{*}$ then $y \in C_1^{*}$. Therefore $C_2^{*} \subset C_1^{*}$