The problem I am trying to solve is mentioned as such:
Let a random variable $ξ$ take values in $Z_+$ and $Eξ < ∞$.
(a) Prove that $ϕ_ξ$ is monotone, bounded and continuous on $[0, 1]$.
(b) Let $ξ_n$ take values in $Z_+$ and $ξ_n \overset{d}{\to} ξ$. Prove that the convergence ${ϕ_ξ}_n\xrightarrow{} ϕ_ξ$ is uniform on $[0, 1]$.
Here $ϕ_ξ$ is generating function for the random variable $ξ$ So from the definition of generating function $ϕ_ξ(Z)$ will be $E Z^ξ$. But I am not sure how to prove the above statements. While searching more on this I found about continuity theorem but I did not find how it is proved. Any lead or help is very much appreciated.
Thanks for the help!!