1 dimensional CW complex, every loop homotopic to finite sequence of edges traversed monotonically I'm having some problems with the following exercise from Hatcher's Algebraic Topology (1.1.19 at page 39):
Show that if $X$ is a path-connected $1$-dimensional CW complex with basepoint $x_0$ a $0$-cell, then every loop in $X$ is homotopic to a loop consisting of a finite sequence of edges traversed monotonically.
First of all I'm not sure what "monotonically" means in this context. Does it mean that you don't go back to a edge or what?
My other problem (besides actually doing the proof) is this: Say X has one 0-cell and an infinite amount of 1-cells, all connected to the 0-cell in both ends. Then how can you find a loop traversing a finite number of edges that is homotopic to the loop traversing all the edges in this $X$ once?
 A: We first show that there exists a partition $0=s_0<\cdots<s_m=1$ such that every closed interval $[s_i,s_{i+1}]$ is mapped to the closure of a 1-cell.
Let $\{x_\alpha\}$ be the set of vertices on the loop. It suffices to show that this is finite. Fix $\alpha$. Denote by $\{E_i\}_{i\in I}$ the set of 1-cells whose closure contains $x_\alpha$. If the closure of $E_i$ contains another vertex, remove it and call the resulting subspace $J_i$. Let $U_\alpha$ be the union of the $J_i$. This is an open neighborhood of $x_\alpha$. Now by construction the $U_\alpha$ form an open cover for the loop, hence compactness yields the conclusion.
Now let $\gamma$ be our loop, and let $\gamma_i:[s_i,s_{i+1}]\to\overline{E_\beta}$, where $\overline{E_\beta}$ is the closure of a 1-cell, be the restriction of the loop to $[s_i,s_{i+1}]$. There are two cases:
1) $\gamma_i(s_i)=\gamma_i(s_{i+1})$. If $\overline{E_\beta}$ has two different vertices, $\gamma_i$ is contractible since $\overline{E_\beta}$ is simply connected. If it has one, it is homeomorphic to $S^1$, so $\gamma_i$ is homotopic to a path traversing $\overline{E_\beta}$ monotonically a finite number of times.
2) $\gamma_i(s_i)\neq\gamma_i(s_{i+1})$. Let $f:[s_i,s_{i+1}]\to\overline{E_\beta}$ be a monotone path from $\gamma_i(s_i)$ to $\gamma_i(s_{i+1})$. Then $\gamma_i\cdot \overline{f}$ is a loop at $\gamma_i(s_i)$ and contractible since $\overline{E_\beta}$ is simply connected. It follows that $\gamma_i$ is path homotopic to $f$.
By combining all the homotopies for segments of $I$, we get a homotopy between $\gamma$ and a loop consisting of a finite sequence of edges traversed monotonically.
