Let $X$ be a random variable with $\mathbb P(X\in\{0,1,2,\ldots\})=1$ The probability generating function of $X$ is defined by $$ \rho(s) = \mathbb E[s^X] = \sum_{k=0}^\infty \mathbb P(X=k)s^k+\begin{cases} 0,& \text{ if } 0\leqslant s <1\\ \mathbb P(X=\infty),& \text{ if } s=1. \end{cases} $$ The claim in question is: $$ \mathbb E \left[\prod_{m=0}^{n-1}(X-m)\right] = \rho^{(n)}(1), \ n=1,2,\ldots $$ We can assume that $\lim_{s\uparrow 1}\rho(s)=1$, as otherwise the claim is trivial. By definition, $$ \rho'(1) = \lim_{s\uparrow 1}\frac{1-\rho(s)}{1-s} = \lim_{s\uparrow 1} \mathbb E\left[\frac{1-s^X}{1-s}\right]. $$ What I don't understand is the following:
As $s$ increases to $1$ through an arbitrary sequence, the nonnegative random variables $(1-s^X)/(1-s)$ form an increasing sequence whose limit is $X$.
The author gives no justification for this claim, and I don't see how it follows. Could I get a hint as to why this is true?