= Radial continuation proof of Hopf-Rinow
Assume <geodesic completeness>. Minimize the distance to a target over a small <compact> <geodesic sphere> about the initial point. The <distance splitting through a small geodesic sphere> gives an initial direction satisfying $d(p,\gamma(t))+d(\gamma(t),q)=d(p,q)$. If this equality stops before the target, repeat the splitting at the last equality point. The resulting broken path minimizes, so its corner must smooth into the same continuing <geodesic>, a contradiction. This supplies a <minimizing geodesic> to every target. Every closed metric ball is then the image under the <Riemannian exponential map> of a closed tangent ball, hence <compact>. A <Cauchy sequence> has a convergent subsequence in such a ball and therefore converges, proving metric completeness without presupposing a global minimizing <curve>.
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