How to prove that a number is irrational

irrational-numbers

We write all postive whole integers after the comma, how do we prove that this is an irrational number? ($0.1234567891011121314...$)


The decimal expansion we were given is not ultimately periodic, for it has arbitrarily long sequences of $5$'s (or any other digit).

But any rational number has ultimately periodic decimal expansion.

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