The sum of multiplicities of two singular points of an irreducible curve of degree n is at most n.

Question

Prove

1. that the sum of multiplicities of two singular points of an irreducible curve of degree $n$ is at most $n$,
2. sum of multiplicities of any 5 points is at most $2n$.

I tried to use Bezuot theorem.

1. Let $P$ and $Q$ are points with multiplicities $p<n$ and $q<n$, then the line $L$ intersects the curve $C$ at least at $p+q < 2n$. As the curve C is irreducible, this number is $n$.

2. I guess The Bezuot theorem is useful here but I have no idea how to applay it.

Could someone check the first proof and correct it if it's wrong and give me idea how to prove the second?