I am studying Riemannian Geometry following my professor's notes. On the proof of existence of affine connection on a $C^\infty$ manifold, the notes states:
By partition of unity, a connection can be locally induced from standard connection on $\mathbb{R}^n$.
I understand the standard connection on $\mathbb{R}^n$, which is a generalization of directional derivative at a point. But how can it induce a connection on the manifold?
The key point is that a finite convex combination of connections on $M$ is itself a connection. (It's easy to check this by hand -- that the combination is convex is what gives you the Leibniz rule.)
So, we cover $M$ by a countable set of charts $\lbrace (\psi_i, U_i) \rbrace$ such that each point $p \in M$ has a neighborhood intersecting only finitely many of the $U_i$. We let $\lbrace \varphi_i \rbrace$ be a partition of unity subordinate to $\lbrace U_i \rbrace$.
On each $U_i$ there exists a connection $\nabla_i$ obtained from the standard connection on $\mathbb{R}^n$ and the chart map $\psi_i$. (This is the connection determined by $(\nabla_i)_{\partial_j} \partial_k = 0$ whenever $\partial_j, \partial_k$ are coordinate vector fields coming from $\psi_i$.)
It follows that the map $$ \nabla = \sum_i \varphi_i \nabla_i, $$ which is a finite convex combination of connections in a neighborhood of any point of $M$, is itself a connection on $M$.