Connected manifold I need to prove a rather simple fact: let $X$ be a connected smooth (or just topological) manifold. Now we need to prove that $X$ is linearly connected.
So, it seems to be obvious, because $X$ is locally like a unit open ball in $\mathbb{R}^n$, but i cannot get clear proof.

You have the right idea. Just show that that being linearly connected is an open condition (i.e. every point has a neighborhood that is linearly connected). Then the linearly connected components' of the space form a decomposition into disjoint open sets. Here thelinearly connected components' are the maximal subsets that are linearly connected.
