I'm self-studying algebraic topology, and would like help with this basic exercise.
My observation is that, the condition $d(f(x),g(x)) < 2$ means that f(x) and g(x) are not antipodal points. Also, following the definition of homotopy, I think my aim is to find a continuous function $H: X \times [0,1] \to S^n$, such that $H(x,0) = f(x)$ and $H(x,1) = g(x)$. But apart from these remarks I don't know how to continue. How should I think about this problem?
I think the usual linear interpolation trick works here:
let H(x,t) = $(tf(x) + (1-t)g(x))/ ||tf(x) + (1-t)g(x)||$
Need to check that H is well defined(obviously H stays on the sphere), i.e. that its denominator is never 0.
We'd like to try H(x,t) = $tf(x) + (1-t)g(x)$ for t $ \in [0,1]$ however this H doesn't stay on the unit sphere(and that's why we define it as above). But since d(f(x),g(x)) < 2, I claim that $tf(x) + (1-t)g(x)$ is never equal to 0. Obviously this holds if $t \neq \frac{1}{2} $ simply because $ |tf(x)|=t \neq (1-t) = |(1-t)g(x)|$ , while for t = $\frac{1}{2}$ if tf(x) + (1-t)g(x) = 0 then f(x) + g(x) = 0 and so d(f(x),g(x)) = |f(x) - g(x)| = |2f(x)| = 2, contradiction.