# Proving a Compact Hausdorff manifold can be embedded in $\mathbb{R}^q$

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.

• Choose a charts around each point $$\{(\varphi_p,U_p):p\in M\}$$, with $$p\in U_p$$, translated and scaled so that $$\varphi_p(p)=0$$ and $$D(2)\subset\varphi_p(U_p)$$.
• Let $$V_p=\varphi_p^{-1}(D(1))$$. Note that the set $$\{V_p:p\in M\}$$ is an open cover of $$M$$
• Choose a finite subcover $$V_{p_1},\dots,V_{p_m}$$ of this cover.
The charts $$\{(\varphi_{p_i},U_{p_m}):i\in [1,m]\}$$ have the desired properties.