The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph

Given: The hypotheses of the Statement.

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

Word length is defined on every element and satisfies $\mathrm{\mid }gh{\mathrm{\mid }}_S\le \mathrm{\mid }g{\mathrm{\mid }}_S+\mathrm{\mid }h{\mathrm{\mid }}_S$, $\mathrm{\mid }{g}^{-1}{\mathrm{\mid }}_S=\mathrm{\mid }g{\mathrm{\mid }}_S$, and $\mathrm{\mid }g{\mathrm{\mid }}_S=0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

A metric on a set satisfies separation, symmetry and the triangle inequality (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric).

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

A Cayley graph is connected if and only if its defining subset generates the group (A Cayley graph is connected if and only if the subset generates the group).

The path metric of a connected simple graph assigns to two vertices the least length of a path joining them (The path metric of a connected simple graph).

The path metric of a connected simple graph is a metric on its vertex set (The path metric of a connected simple graph is a metric on its vertex set).