  In our previous work with Grifo and H\`a, we showed the stable
Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal
of sufficiently many general points in $\mathbb{P}^N$. In this paper, we
establish the conjectures for all remaining cases, and hence, give the
affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture
for any number of general points in $\mathbb{P}^N$ for all $N$. Our new
technique is to develop the Cremona reduction process that provides effective
lower bounds for the Waldschmidt constant of the defining ideals of generic
points in projective spaces.
