Give a proof that shows where =0,1,2,3... is irrational.
The idea here is to show that the decimal expansion of x does not terminate or eventually repeat.
Assume that this number, call it $\displaystyle x$, is rational, then $\displaystyle x=\frac{a}{b}$. And thus by this it means we can write $\displaystyle \frac{a}{b} = \sum_{k=0}^{\infty} b_k 10^{-k}$ where $\displaystyle \{ b_k\}$ is eventually repeating. Therefore, $\displaystyle x - \frac{a}{b} = 0 = \sum_{k=0}^{\infty} (a_k - b_k)10^{-k}$. But the only way to express zero in a decimal expansion is trivially i.e. $\displaystyle a_k - b_k = 0 \implies a_k = b_k$. But then this implies that $\displaystyle \{a_k\}$ must also be eventually repeating which is a contradiction because it is an increasing sequence. Contradiction.