If $F : M^k \to S^n$ is a smooth map, $M$ compact, $n > k$, then $F$ is homotopic to a constant map

Question

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map.

Proof: Since $n>k$, by Sard's theorem, the image of $M$ is nowhere dense, hence not surjective. Pick a point not in the image. Using the Riemann projection, we have a homeomorphism $G$ from $S^{n}-\{x\}$ to $\mathbb{R}$, which is convex, hence there is a homotopy between $G\circ F$ and the constant map. Composing the homotopy with $G^{-1}$ we have the desired homotopy in $S^{n}$.

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Answer 1

Your answer is correct, yes.${}$