i wish to show that open cubes centered at every point form a basis for $R^{n}.$ In this, i am trying to show than an open ball is a union of smaller open cubes, in which case is open, and hence the cube metric forms a basis.
Now, obviously, i can find an open cube inside every open ball, and i can take a union of them to cover the ball, which will form an open set, but how can i show that i can form EXACTLY a ball with such a union?
All you need to do is this: for every point in the open ball, take an open cube that contains that point and is a subset of the ball. A moment's thought should show: (i) the union of the cubes is a subset of the ball; and (ii) since every point in the ball belongs to a cube, the ball is a subset of the union of the cubes. If two sets are subsets of each other, they are equal.