It is widely thought that $\pi$ contains every possible finite digit sequence. But what if the sequence is infinite? When the first digit of the sequence appears somewhere in $\pi$ it should never end since both $\pi$ and the sequence are infinite. Does that mean that the sequence itself contains every possible sequence since it's essentially $\pi$ minus the digits from the first digit of pi to the digit where the sequence's first digit appears?
What about pi itself? Of course, I'm not talking about the first digit starting from first digit of $\pi$. That's kind of obvious. If I were to find $\pi$ in $\pi$ digits would that mean that $\pi$ actually is repeating? I guess that's impossible because that would mean $\pi$ is actually a rational number which it isn't. Am I thinking correctly here?
So what about infinite sequences in $\pi$? Any thoughts?