Show that the decimal number $0.12624120720...$ obtained by concatenating the digits of $n!$ successively with $n = 1, 2, 3, ...$ represents an irrational number.
A rational number either has a terminating decimal expansion or an eventually repeating decimal expansion. $0.12624120720... $ is clearly not a terminating sequence. $n!$ is unique for each $n \in \mathbb N$ But how do we prove that uniqueness means non-repeating here?
Hint: can you prove that there are arbitrarily long sequences of zeroes - eventually longer than any claimed finite period?