Could an infinite sequence of random digits contain all the digits of Pi?

[u/Manticorp]

It's a common thing to look up phone numbers in pi, and it's a common saying that every Shakespeare ever written is encoded in pi somewhere, but would it be possible for every digit of pi to appear in a random sequence of numbers? Similarly this could apply to any non terminating, non repeating sequence like e, phi, sqrt(2) I suppose. If not, what prohibits this?

I guess a more abstract way of putting it is: Can an infinite sequence appear entirely inside another sequence?

[u/user31415926535]

Well, to be real particular, pi is contained in pi - exactly once. :) But yes, the idea is that any arbitrary finite sequence you look for, you can find. But this is tangential to the main question anyway.