Monotone Convergence Theorem of random variables is stated as below:
Assume that there is a sequence of random variables (r.v.) satisfing $0\leq X_1\leq X_2 \leq \cdots$ and $X_n\to X$ a.s.(almost surely), then $$E[X_n]\to E[X]$$ where E is expectation.
I read about a example using the Monotone Convergence Theorem on the text book and found some problems. The example is stated as follows:
Assume r.v. X is non-negative, and denote $\mu=EX$, define sequence of r.v.: $$X_n=\min\{X,n\},n\in \mathbb{N}$$ Then as $n\to\infty$, $EX_n=\mu$ because $X_n$ is monotone.
My question is, why they say $X_n$ is monotone when there are posotive probability of $X_{k-1}>X_{k}$? For example, as $X_n$ are independent, $$P(X_{k-1}= k-1, X_k<k-1)=P(X_{k-1}= k-1)P(X_k<k-1)$$ $$=P(X\geq k-1)P( X<k-1)>0.$$