I had to ask myself two simple problems in differential topology: 1) Why is the rank of a diffeomorphism (on a manifold of dimension m) of rank m? 2) Why is a chart on a manifold an embedding? These are actually quite obvious so textbooks don't even bother proving it. So I've attached my own proofs (note that the Inverse Function Theorem goes only one way, so I couldn't use it to prove #1). My proofs probably seem longer than they need to be, but I wanted to be completely rigourous. Are they correct?