Homotopic maps between spheres

Question

I have read somewhere that two maps $f,g:S^n\rightarrow S^n$ satisfying

$$ |f(x)-g(x)|<2 \qquad \forall \ x\in S^n $$

are homotopic. How can one show this (or does someone have a reference)? I have no idea where to even start with such a statement...

Answer 1

Consider the map $H:S^n\times I\to S^n$ such that $H(x,t)=\frac{tf(x)+(1-t)g(x)}{\lVert tf(x)+(1-t)g(x)\rVert}$. Use your hypothesis to show that this is welll-defined, that is, that the denominator never vanishes.