Show that if $f:S^8 \to S^8$ and $\|f(x) + x\| < 1$ for all $x$ then $f$ is not homotopic to the identity map

Question

I need to show that if a map $f:S^8 \to S^8$ satisfies $\|f(x) + x\| < 1$ for all $x \in S^8$ then $f$ is not homotopic to the identity map.

Progress

I saw some two related propositions:

• Proposition 1: Antipodal map $f: S^{n} \rightarrow S^{n}$ has degree $\deg(f) = (-1)^{n+1}$.
• Proposition 2: If $n$ is even than $f$ is not homotopic to identity.

Is this related? I also not sure, how can I show that $f$ in my question is homotopic to antipodal map?