My attempt to this problem is as follows:

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.

Let

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 .