To show is a basis, we need to show that
(1) For each , there is at least one basis element containing x.
Let x be . Then there is a basis element containing x such that , n is a positive integer.
(2) If x belongs to the intersection of two basis elements and and then there is a basis element containg x such that .
There are several cases of intersections, we find the basis element satisfying the above.
For instance, and , and if x belongs to the above , then x belongs to the below
In each cases, can be described as
Thus, is a basis for .