So let's start by setting . Now for we can take but in order to keep it disjoint from we take away . - So and are disjoint, agreed?
Similarly, for we can take , but in order to keep it disjoint from must take away their union, or, what amounts to the same thing . We choose .
So generally, we would like to define .
Observe that , for all i, by definition. Also, if we see that
Finally, to show that .
First, from for all i it follows immediately that
Second, suppose that . In that case there must exist a smallest , such that .
But then we have that and therefore , a contradiction. Thus there can be no such x, hence we have that