This is a 9GAG picture I saw tonight. The way it's put, it is evidently false, since 0.10100100010000… (the powers of 10 all in a row) is definitely decimal, infinite and nonrepeating (or in one word, irrational), but most surely doesn't contain every possible number combination. I was just wondering: can this be proved for pi? And more in general, are there sufficient conditions for a decimal number to contain, in its decimal expansion, all possible number sequences?
PS I wasn't sure how to tag this. I think this question deserves more than just the tag "pi". Any ideas on another tag for this question?
It is not known which patterns of numbers occur in the decimal expansion of $\pi$. I saw a reference that said we don't even know whether or not arbitrarily long strings of $9$'s, for example, occur.