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I recently have started learning some elementary homotopy theory, and I'm having some trouble proving this statement. Over here a "polygonal path" means a path consisting of a finite number of points $s_1 = 0 < s_2 < ...< s_n = 1$ and for our path $p$, for every $i$, $p(s_i)$ is joined by $p(s_{i+1})$ by some arc of a geodesic of $S^n$, i.e an arc of a great circle. How should I proceed?

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