A containment result in $\mathbb{P}^n$ and the Chudnovsky conjecture

In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ points in $\mathbb{P}^n$.