Hopf–Rinow theorem

(i) $\Rightarrow$ existence of a minimizing geodesic: fix $p,q \in M$ and let $r = d(p,q)$. The small metric sphere $S_\varepsilon(p)$ is compact, so the continuous function $x \mapsto d(x,q)$ attains a minimum on it at some point $x_0 = \gamma(\varepsilon)$, where $\gamma$ is the unit-speed geodesic from $p$ through $x_0$ (defined for all time by geodesic completeness). One shows $d(\gamma(\varepsilon),q) = r-\varepsilon$, and then extends this to the set $A = \{t \in [0,r] : d(\gamma(t),q) = r-t\}$: $A$ is nonempty and closed by continuity, and repeating the same minimizing-sphere argument at $\gamma(t_0)$ for any $t_0 \in A$ with $t_0<r$ shows $A$ is also open in $[0,r]$, so $A=[0,r]$ and $\gamma(r)=q$.

(i) $\Rightarrow$ (ii): given a Cauchy sequence $(x_n)$ in $M$, the argument above shows any two points sufficiently close are joined by a minimizing geodesic whose length equals the distance between them; a closed metric ball around any fixed point is then the continuous image of a closed ball in $T_pM$ under the exponential map, hence compact, so the Cauchy sequence eventually lies in a compact set and converges.

(ii) $\Rightarrow$ (iii): metric completeness together with local compactness (every Riemannian manifold is locally compact) implies that closed balls are totally bounded and complete, hence compact by the standard metric-space characterization of compactness; a general closed and bounded set is a closed subset of some closed ball, hence compact as a closed subset of a compact set.

(iii) $\Rightarrow$ (i): suppose a unit-speed geodesic $\gamma$ is only defined on a maximal interval $[0,T)$ with $T<\infty$. As $t \to T^-$, $\gamma(t)$ stays within the closed ball of radius $T$ about $\gamma(0)$, which is compact by (iii); a sequence $\gamma(t_n)$ with $t_n \to T$ then has a convergent subsequence, and standard existence theory for the geodesic ODE (a second-order system with smooth coefficients) shows the solution extends smoothly past $T$, contradicting maximality of $T$. Hence every geodesic extends to all of $\mathbb{R}$.