  In this paper, a normality criterion concerning a sequence of meromorphic
functions and their differential polynomials is obtained. Precisely, we have
proved: Let $\left\{f_j\right\}$ be a sequence of meromorphic functions in the
open unit disk $\mathbb{D}$ such that, for each $j,$ $f_j$ has poles of
multiplicity at least $m,~m\in\mathbb{N}.$ Let $\left\{h_j\right\}$ be a
sequence of holomorphic functions in $\mathbb{D}$ such that $h_j\rightarrow h$
locally uniformly in $\mathbb{D},$ where $h$ is holomorphic in $\mathbb{D}$ and
$h\not\equiv 0.$ Let $Q[f_j]$ be a differential polynomial of $f_j$ having
degree $\lambda_Q$ and weight $\mu_Q.$ If, for each $j,$ $f_j(z)\neq 0$ and
$Q[f_j]-h_j$ has at most $\mu_Q + \lambda_Q(m-1)-1$ zeros, ignoring
multiplicities, in $\mathbb{D},$ then $\left\{f_j\right\}$ is normal in
$\mathbb{D}.$
