I have some trouble with the following problem:
"Let $X,X_1,X_2,...$ be non-negative integer-valued random variables. Show that if $X_n \overset{d}{\to} X$ and X is almost surely finite, then the probability generating functions converge pointwise on $[0,1]$, that is $g_{X_n}(t) \to g_X(t)$ for $t\in[0,1]$."
I feel confused with the sentence "X is almost surely finite", do they mean that $X\overset{a.s}{\to}c$ if $c<\infty$?
I would be most grateful if someone could translate that sentence and at least give me a hint on how to start solving this problem.
$X_n \overset{d} \rightarrow X$ means that for all continuous and bounded functions $f$ we have $$\Bbb E [f(X_n)] \rightarrow \Bbb E [f(X)].$$ Fix $t\in[0,1]$. Choose $f:\Bbb R \to \Bbb R, x \mapsto t^x$. Therefore, $$g_{X_n} (t) = \Bbb E [t^{X_n}] \to \Bbb E [t^{X}] = g_X (t).$$