My problem is the following:
Let $G$ be a group with generating set $X$. We can look the Cayley-Graph $\Gamma(G,X)$ of $G$. Let $x\in G$. Then it holds: $d_{\Gamma}(v,xv)\leq 1$ for all $v\in G=\Gamma(G,X)$ if and only if $x\in X$. Why is that true?
I know that $d_{\Gamma}(v,vx)=d_X(v,vx)=|v^{-1}vx|_X=1$, where $d_X$ is the word metric on $G$ relative to X, iff $x$ lies in $X$.
I think its not very difficult, but I think I make a mistake in my thinking about the problem.
Thanks for help.