  Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$
over an algebraically closed field $F$. For each positive integer $m$, let
$\alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in
$F[x_0,\ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$.
The Waldschmidt constant $\widehat{\alpha}(Z)$ of $Z$ is defined by the limit
\[
  \widehat{\alpha}(Z)=\lim_{m \to \infty}\frac{\alpha(mZ)}{m}. \] Demailly
conjectured that \[ \widehat{\alpha}(Z)\geq\frac{\alpha(mZ)+n-1}{m+n-1}. \]
Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when
$Z$ is very general and \[
  \lfloor\sqrt[n]{s}\rfloor-2\geq m-1. \] Here we improve their result and show
that Demailly's conjecture holds if $Z$ is very general and \[
\lfloor\sqrt[n]{s}\rfloor-2\ge \frac{2\varepsilon}{n-1}(m-1), \] where $0\le
\varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$
very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$),
Demailly's conjecture holds for all $m\in\mathbb{N}$. We also show that
Demailly's conjecture holds if $Z$ is very general and \[
  s\ge\max\{n+7,2^n\}, \] assuming the Nagata-Iarrobino conjecture
$\widehat{\alpha}(Z)\ge\sqrt[n]{s}$.
