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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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Suppose $\gamma:[0,1]\to S^n$ is a continuous map. Let $\mathcal U$ be the open covering of $S^n$ by the eight open half hemispheres.

There is a positive integer $m$ such that for all $i\in\{0,\dots,m-1\}$ the image of the interval $[\tfrac{i}{m},\tfrac{i+1}m]$ under $\gamma$ is completely contained in one of the open sets in $\mathcal U$, which I will call $U_i$. This is a consequence of the fact that $[0,1]$ is a compact metric space, that $\{\gamma^{-1}(U):U\in\mathcal U\}$ is an open covering of $[0,1]$, and of the Lebesgue's number lemma.

Let now $\sigma:[0,1]\to S^n$ be the polygonal map such that

  • $\sigma(\tfrac{i}{m})=\gamma(\tfrac{i}{m})$ for all $i\in\{0,\dots,m\}$, and

  • for all $i\in\{0,\dots,m-1\}$, the restriction $\sigma:[\tfrac{i}{m},\tfrac{i+1}{m}]\to S^n$ is a geodesic, with image completely contained in $U_i$.

I leave it as an exercise for the OP to prove that $\sigma$ and $\gamma$ are homotopic paths (relative to their endpoints, even)

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