One of the exercises in the text I'm using for self-study asks to prove that the union of a pair of atlases A and B on a manifold is another atlas. However, I don't see any way to show that two charts C,D in [tex]A\cup B[/tex] with [tex]C\in A~,~D\in B[/tex] are compatible. Could anyone give me a bit of help? Maybe just a hint? The book is A Course in Modern Mathematical Physics by Peter Szekeres. The exercise is the first on in chapter 15.