Can it be proven/disproven that every possible number combination is in Pi?

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Pi is infinite, but not random. 1/3rd for example is certain not to contain a 4, but i found my full birthday in Pi. Is there proof that every conceivable combination of numbers, such as all Natural numbers in order? I'm having a hard time imagining how all natural numbers could have something before and after them though so can it be proven that every finite numbercombination is found in Pi, such as every birthday of every person in chronological order and everyones creditcard pin and then some. Or is it likely/unlikely/impossible(or impossible to tell)? I'm asking out of my own curiosity.