Ask HN: Does an irrational, normal number generate all strings?

Recently the Everything2 article "Converting Pi to binary: Don’t do it!" [1] has been a popular read on Reddit. It begins, "WARNING: Do NOT calculate Pi in binary. It is conjectured that this number is normal, meaning that it contains ALL finite bit strings." Now all that a number being "normal" means is, all the digits in its base tend to show up in roughly equal proportions. My question is whether the following is valid reasoning, and why or why not: Given: 1. Pi is irrational. 2. Pi is (assumed to be) normal. Conclude: 3. For any string consisting of digits from the base Pi is written in, this string exists somewhere in Pi. It seems counter-intuitive in its broadness, but if Pi is infinite and never repeats, then I guess we have a classic infinite-typing-monkey situation, right? At the same time, this conclusion would mean we should expect to eventually find a string of a googol consecutive zeroes within the digits of Pi. [1] http://everything2.com/title/Converting+Pi+to+binary%253A+Don%2527t+do+it%2521