When does the cardinality disappear? Pardon me, if this question sounds stupid. I am learning real analysis on my own and stumbled on this contradiction while reading this -- http://math.kennesaw.edu/~plaval/math4381/setseq.pdf. I appreciate any help or pointers.
Consider this example:  We choose an open interval of $(0, \frac{1}{n})$ for each n from N, the set of natural numbers i.e. N = {1, 2, 3, ...}. Let us call this interval I(n) as in the pdf.
Now, let us form intersections of all the I(n) we have at any point. I(1) is (0, 1), which has the cardinality of R. Then we form I(2) and look for intersection between I(1) and I(2), which is I(2). When I use n = 3, I get I(3) and the intersection of I(1), I(2), and I(3) is I(3).
But, I can transform $(0, \frac{1}{2})$ and then map it back to R. So |I(2)| = |I(1)| -- that is, the cardinalities of I(1) and I(2) are the same.
Now, I can repeat this argument for each n. Thus, |I(n)| = |I(1)| for all n in N. Thus, the intersection of {I(k)}, k in {1, 2, 3, ..., k} has the cardinality of R.
But, as $n \to \infty$, the intersection is the empty set, whose cardinality is 0.
So, I see that the cardinality of intersections of I(n) went from R to 0. 
But at every step, the cardinality does not change. Nor do I see it decrease.

My question then is -- how does the cardinality simply disappear when I don't see it changing at any step? Did I make one or more wrong assumptions?


In essence this boils down to a question about the interchange of the limit and the cardinality function. In general it is not true that
$$
\lim_{n\to\infty} card(C_n) = card\left(\lim_{n\to\infty} C_n\right)
$$
