Geodesic completeness and complete Killing fields

The corresponding flow, say $\Phi^t: M\to M$ preserves the metric and the field. Thus, for any $x\in M$, the curve $\alpha_x\colon t\mapsto \Phi^t(x)$ has constant speed. Therefore it can not escape to infinity in finite time.

More precisely: if $\alpha_x$ is defined on a bounded interval $(a,b)$ then the restriction $\alpha_x|(a,b)$ has finite length, and from completeness it can be extended to a neighborhood of $[a,b]$. This implies that $\alpha_x$ is defined on whole $\mathbb R$; i.e., the vector field is complete.