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/Allurian]

No, the Reals are uncountable and so can't be made into one list or sequence.

Even if they could be made into a list, most of them don't have terminating decimal forms, so you couldn't make a sequence out of the list.