6. Show that for all $n \in \mathbb{N}$, for every function $f$ of $S_n$ satisfying $\lim_{x \rightarrow \pm\infty} |f(x)| = \pm\infty$, there exists an element $g \in \mathscr{P}_{n+1}$ such that $f \sim g$ (where $\sim$ is the relation defined in I.2).