I have been reading Hirsch's Book Differential Topology and here he claims that
Let $M$ be a compact Hausdorff manifold of class $C^r$, $1\leq r\leq \infty$. Then there exists a $C^r$ embedding of $M$ into $\mathbb{R}^q$ for some $q$.
Now I only have one doubt in this proof that is in the beggining he claims that if we have charts $\{(\phi_i,U_i)\}_{i=1}^m$ that cover $M$, by composing with a diffeomorphism if necessary we can assume that $D(2)\subset \phi_i(U_i)$ and that $M=\cup_{i=1}^m int(\phi_i^{-1}(D(1)))$. Now I get each of the conditions independently but I can't seem to make both work at the same time, especially in the case that we have a chart $U_j$ that is disjoint from every other chart. Any help clarifying this is appreciated. Thanks in advance.