Smooth structures on compact manifolds I am currently reading some notes where an operator, originally defined on functions over Euclidean Space is now transferred to the setting of a compact, smooth Riemannian manifold.
There is a statement in the notes which says the following:
"Let $M$ be a smooth compact Riemannian manifold without boundary. Cover $M$ by a finite number of coordinate charts $U_i$ with diffeomorphisms $h_i : O_i \to U_i$ where the $O_i$ are open subsets of $\mathbb{R}^m$ with compact closure. We assume the coordinate charts $U_i$ are chosen so that the union $U_i \cup U_j$ is also contained in a larger coordinate chart for all $(i,j)$."
This last statement confuses me a little, because I cannot see this as being trivial. How can I justify this assumption ? 
That is, given I have a smooth atlas $\{ U_i \}$ for $M$ (which we can take to consist of finitely many charts since $M$ is compact), how do I adjust this chart so that the last sentence above holds ? 
Many thanks for your help!
 A: I'm interpreting the statement as follows:
There is a finite collection of distinguished coordinate charts $U_i$ covering $M$ where each $U_i\cup U_j$ is contained in another (perhaps nondistinguished) coordinate chart $W_{ij}$.
I am not necessarily claiming that $W_{ij}\cup W_{kl}$ is contained in a chart $Z_{ijkl}$ and each $Z_{ijkl}\cup Z_{mnop}$ is contained in a chart ...., just that the union of the $U$s are contained in charts.
Here's a proof that such things always exist, though it may use tools you don't have access to.
Equip $M$ with any Riemannian metric $g$.  This gives rise to several functions.  First, for each point, there is a map called the exponential map, $\operatorname{exp}_p :T_p M\rightarrow M$ with the property that $\operatorname{exp}_p$ is a diffeomorphism onto its image when restricted to a small enough ball around the origin in $T_p M$.  It is a fact that when $\operatorname{exp}_p$ is restricted to a ball of radius $r$ in $T_p M$, the image consists of all points in $M$ a distance $r$ away from $p$.
Another important function coming from a choice of metric is the function $\operatorname{inj}:M\rightarrow\mathbb{R}$.  It is called the injectivity radius and defined as the largest radius of a ball around the origin in $T_p M$ such that $\operatorname{exp}_p:B(r)\rightarrow M$ is a diffeomorphism onto its image.  It is known that $\operatorname{inj}$ is a continuous function and, on a compact manifold, bounded away from $0$.
Let $\rho$ denote the minimum value of $\operatorname{inj}$ on $M$.  Then, by definition, at every point $p$, if we restrict $\operatorname{exp}_p$ to the ball of radius $\rho$, it's a diffeomorphism onto its image.  In particular, we can use $\operatorname{exp}_p(B(\rho))$ as a chart on $M$.
Let $U_p$ denote $\operatorname{exp}_p(B(\frac{\rho}{2}))$.  I claim that the collection of $\{U_p\}$ has the property that the union of any 2 are contained in a coordinate chart and that they cover $M$.  Once we establish all this, use compactness of $M$ to extract a finite subcollection of the $U_p$, giving the desired collection of charts.
First, $p\in U_p$ since $\operatorname{exp}_p(0) = p$ always.  Thus, these do cover.
Now, why do they have the property that the union is contained in a coordinate chart?  We'll break into 2 cases depending on whether or not $U_p\cap U_q$ is empty or not.
If $U_p\cap U_q =\emptyset$, then $U_p\cup U_q$ is a (disconnected) chart.  The diffeomorphism between it and an open subset of $\mathbb{R}^n$ is given by $$\begin{cases} \operatorname{exp}_p^{-1}(r) & r\in U_p \\ \operatorname{exp}_q^{-1}(r) + v & r\in U_q\end{cases}, $$ where $v$ is some vector of very large (compared to $\rho$) length.  The point of $v$ is to shift the ball $\operatorname{exp}_q^{-1}(U_q)$ in $T_q M\cong \mathbb{R}^n$ far enough away from the origin so that it wont intersect the image of $\operatorname{exp}_p^{-1}(U_p)$ in $T_p M \cong\mathbb{R}^n$
On the other hand, if $U_p\cap U_q \neq \emptyset$, choose $r\in U_p\cap U_q$.  I claim that $U_p\cup U_q \subseteq \operatorname{exp}_r(B(\rho))$.  To see this, we'll use the triangle inequality.  Using $d$ to denote distance, we have for any $s\in U_p$, that $d(s,r)\leq d(s,p) + d(p,r) < \frac{\rho}{2} + \frac{\rho}{2} = \rho$.  This same argument works for $U_q$, so we have $U_p\cup U_q\subseteq \operatorname{exp}_r(B(\rho))$.
