  We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$,
where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that
$f_n$ converges uniformly to a map $f$. We show, among others, that if $f$ is
chaotic in the sense of Li and Yorke then the nonautonomous system $ \{
f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and that the same is true for
distributional chaos. If $f$ has zero topological entropy then the
nonautonomous system inherits its infinite $\omega$-limit sets.
