  We prove Demailly's Conjecture concerning the lower bound for the Waldschmidt
constant in terms of the initial degree of the second symbolic powers for any
set of generic points or very general points in $\mathbb{P}^N$. We also discuss
the Harbourne-Huneke Containment and the aforementioned Demailly's Conjecture
for general points and show the results for sufficiently many general points
and general points in projective spaces with low dimensions.
