Covering a smooth manifold with a sequence of charts Let $M$ be a connected smooth manifold. For simplicity, I introduce the following definition:

A sequence $(U_n)_{n\in\mathbb{N}}$ of connected charts of $M$ is good if $U_i\cap U_{i+1} \neq \varnothing$ for all $i\in\mathbb{N}$.

Intuitively, I think that for any connected chart $U$ there is a good sequence starting with $U$ that covers the whole manifold, but I don't know how to (dis)prove this.
I'd really like to show some progress or ideas but it feels so intuitively true that I don't know where to begin...
 A: Let ${\mathcal U}$ be a countable open cover of $M$ by coordinate charts (it does not matter exactly what you require from these, as long as they are open in $M$). Let $N$ denote the nerve of this cover and $G$ be the graph of the 1-dimensional skeleton on $N$. In other words, vertices of $G$ are elements of ${\mathcal U}$ and two distinct vertices are connected by an edge if and only if the two elements of ${\mathcal U}$ have nonempty intersection.
Lemma 1. $G$ is connected.
Proof. If not, then the vertex set of $G$ can be partitioned in two nonempty subsets $V_1, V_2$ such that no element of $V_1$ is connected to an element of $V_2$ by an edge. Let $M_i$ denote the union of all open subsets which are vertices in $V_i$, $i=1,2$. Then $M_1\cap M_2=\emptyset$ and $M_1\cup M_2=M$. Since $M$ is connected, this is a contradiction. Hence, $G$ is connected. qed
The next lemma is a simple exercise in combinatorics:
Lemma 2. Suppose that $G$ is a connected simple graph with countably many vertices. Then given any vertex $v$ in $G$, there exists an infinite combinatorial path $p$ in $G$ starting at $v$ and passing through every vertex of $G$.
Now, apply Lemma 2 to the graph $G$ from Lemma 1 where the initial vertex $v$ corresponds to the given element $U\in {\mathcal U}$. Let $U_i\in {\mathcal U}$ denote the $i$-th vertex in the path $p$ in the graph $G$. Then, by the construction, $U_i\cap U_{i+1}\ne \emptyset$ and, hence, the sequence $(U_i)_{i\ge 0}$ is a good sequence of charts covering the entire $M$.
