Let $(M,g)$ be a complete Riemannian manifold, let $d_g: M \times M \to [0,\infty)$ denote its Riemannian distance, and let $\rho: [0,\infty) \to M$ be a ray, meaning that $\rho$ is a unit-speed geodesic satisfying $d_g(\rho(s),\rho(t)) = |t-s|$ for all $s,t \ge 0$. Define the Busemann function associated to $\rho$ by
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\begin{align*}
b_\rho: M &\to \mathbb{R} \\
x &\mapsto \lim_{t \to \infty} \bigl(t - d_g(x,\rho(t))\bigr).
\end{align*}
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Then the defining limit exists for every $x \in M$, the function $b_\rho$ is $1$-Lipschitz with respect to $d_g$, and