Show that if $\{P_n\}$,$ n\geq0$ is orthogonal with respect to a linear functional $L$ then the following two are equivalent.

Question

Let $\{P_n\}$,$ n\geq0$ be a polynomial sequence such that $deg (P_n)=n$ for all $n ≥ 0$. Show that if ${(P_n)}_{n≥0}$ is orthogonal with respect to a linear functional $L$ then

$L[f_m(x)P_n(x)] = 0$ for any polynomial $f_m(x)$ of degree $m<n$,

$L[f_n(x)P_n(x)] = 0$ for any polynomial $f_n(x)$ of degree $n≥0$.

are equivalent.

I have no idea where to begin with this proof, any help would be greatly appreciated.