An infinite finitely generated group contains an isometric copy of $\mathbb{R}$, i.e., contains a bi-infinite geodesic The question is: prove that an infinite finitely generated group $G$ contains an isometric copy of $\mathbb{R}$, i.e., contains a bi-infinite geodesic ($G$ is equipped with the word metric).
I do not even know what I have to prove. It does not make sense to me. The word metric of $G$ assumes values in the natural numbers. How could there be an isometry between a subgraph of the Cayley graph of $G$ and the real line $\mathbb{R}$.
I am really confused.
I found this question here (sheet 6, ex. 1).

I'm just going to focus on what you've said you are confused about, namely:
"How could there be an isometry between a subgraph of the Cayley graph of G and the real line $\mathbb{R}$?".
We can extend the word metric on $G$ to a metric on the Cayley graph in a natural way, with each edge being an isometric copy of a unit interval.  Under this metric, the Cayley graph of $\mathbb{Z}$ with respect to the generator $1$ is isometric to $\mathbb{R}$.
