Proof:

Suppose $f({\bf x}) \neq {\bf x}$ for any ${\bf x} \in S^n$, we see by theorem (36.8) that $f$ is homotopic to the antipodal map. Further, if also $f({\bf x}) \neq -{\bf x}$ for every ${\bf x} \in S^n$, theorem (36.8) implies $f$ is homotopic to the identity map. This contradicts corollary (36.7).