I have problems with the proof why every manifold admits a Riemannian metric. If the manifold $M$ has an atlas given by sets $U_i$, we can take a subordinate partition of unity $\{\phi_i\}_i$. Then $\sum \phi_i g_i$, where $g_i$ is the standard metric on $U_i \tilde = \mathbb R^n$, defines a riemannian metric.
I don't understand why this is well defined. Isn't there some transformation condition we need to to check when switching charts on their overlap?