Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
is not rational.
Since we know that there are infinite natural numbers, the decimal is non-terminating.
Since the decimal is written such that the natural numbers are in increasing order, numbers will not repeat themselves.
Hence 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ... is a non-terminating, non-repeating decimal, hence not a rational.
0. 123 123 123 123 123 ... (Usually expressed as $\displaystyle 0.\overline{123}$) is a rational number because it can be written in terms of $\displaystyle \frac{p}{q}$. How? Using infinite geometric series:
$\displaystyle \text{S} = \frac{t_{1}}{1-\text{r}}$
Where t_1 is the first term in the series and r is the common ratio between each consecutive term.
$\displaystyle 0.\overline{123} = \frac{123}{1000} + \frac{123}{1000000} + \ldots$
$\displaystyle r = \frac{\frac{123}{1000000}}{\frac{123}{1000}} = \frac{1}{1000} $
Now substitute in the equation above:
$\displaystyle \text{S} = \frac{t_{1}}{1-\text{r}} = \frac{\frac{123}{1000}}{1-\frac{1}{1000}} = \frac{\frac{123}{1000}}{\frac{999}{1000}} = \frac{123}{999}$
A rational number has a decimal expansion which is eventually periodic, that is from some point on there is a string of digits of some length that is just repeated continually.
That the given number does not have that property just consider how far to the right the digit strings: 10, 100, 1000, 10000, ... first appear. What this shows is that for any given N the expansion has not become periodic before reaching N digits (since one of these strings first appears to the right of the N'th digit. Hence it does not become periodic period.
RonL