I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample:
$F_n(x)=0$ if $x<n$ and $1$ otherwise. It is a distribution function for any $n$. Now $\lim_{n\to \infty}F_n(x)=0$ for all $x \in R$. The resulting function $F(x)=0$ for all $x \in R$ is not a distribution function anymore.
Would you agree with the counterexample? Do you know other counterexamples?