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 , is rational, then . And thus by this it means we can write where is eventually repeating. Therefore, . But the only way to express zero in a decimal expansion is trivially i.e. . But then this implies that must also be eventually repeating which is a contradiction because it is an increasing sequence. Contradiction.