Is it rational?

Assume that the decimal $0.1234....$ is periodic, that $n$ is the periodicity(number of digits in a period), and that $k$ is the number of digits encountered before the period position starts. Consider the integer $10^{m}$ , where $m$ is not less than $n+k$ . In composing the decimal we wrote in succession all the integers; hence any chosen number $N$ will appear somewhere. Since in the sequence of numbers written in to make up the infinite decimal $m \geq n+k$ zeros must be encountered, it follows that the only possible period consists of one zero- a situation that does not hold for this decimal. Hence the decimal is not periodic.