Proving Compatibility of Charts in a Union of Atlases

[jojo12345]

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 A\cup B with C\in A~,~D\in B 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.

 

[jojo12345]

The wording of the exercise is as follows:

If A and B are two atlases on a manifold M, then their union is another atlas. Prove this statement. [Hint: A differentiable function of a differentiable function is always differentiable]

This is really verbatim from the text. However, after reading the line in the text after the exercise, I get the impression that the author made a typo. The next line:

"Any atlas can this be extended to a maximal atlas by adding to it all charts that are compatible with the charts of the atlas."

To me this suggests two things. First, the proposition in the exercise is false. Second, what the author actually wanted to ask was the following:

Show that if (1) A,B, and C are atlases on M, (2) the charts in A are compatible with the charts in B, and (3) the charts in B are compatible with the charts in C, then the charts in A are compatible with the charts in C (transitivity).

What do people think?