Relation betwen a $p$ norm and dual norm

Question

Consider $$ A=\{y\in \mathbb{R}^{n}: \langle y,x \rangle\le \lVert x \rVert_p \quad \mbox{ for all }x\in \mathbb{R}^n \} $$ and $$ B=\{x\in \mathbb{R}^{n}: \lVert x \rVert_{q}\le 1 \} $$ where $p, q\in [1,\infty]$ such that $\frac{1}{p}+\frac{1}{q}=1$. How can I prove that $A\subseteq B$. I have no idea how to start with this demonstration. Any help will be appreciated.

Answer 1

Hint: Use the vector $x$ with $x_i = \operatorname{sign}(y_i) \, |y_i|^\alpha$ for some appropriate power $\alpha \in \mathbb R$.