I'm a bit confused by the definition of convergence of a sequence of random variables. I'm reading Wasserman's All of Statistics and at the start of Chapter 5, he writes:
"Suppose that $X_1, X_2...$ is a sequence of random variables which are independent and suppose each has a N(0, 1) distribution. Since these all have the same distribution, we are tempted to say that $X_n$ converges to $X$~$N(0,1)$. But this can't quite be right since $P(X_n = X) = 0$ for all $n$. (Two continuous random variables are equal with probability 0.)"
Why is it the case that two continuous random variables are equal with probability 0? It seems that he's using a different definition of "equal" as I would have thought two random variables with the same distribution are equal in some sense...
Any help clearing this up is much appreciated!