In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky’s conjecture for the defining ideal of sufficiently many general points in PN. 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 the general points in PN 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.