Here is problem statement:

Show that if X is any topological space, and if two continuous maps $\displaystyle f,g:X \to \mathbb{S}^n$ have the property that for all $\displaystyle x\in X$, $\displaystyle \| f(x)-g(x)\| < 2 $, then f is (continuously) homotopic to g.

And here is a homotopy (in our case)

A function $\displaystyle F(x,t): X \times I \to \mathbb{S}^n$ such that $\displaystyle F(x,0)=f(x)$ and $\displaystyle F(x,1)=g(x)$.

I don't have any trouble coming up with a function that satisfies the "deforming" property...for instance, there is a smooth function (I won't prove it, but there is, see Milnor or any other diff. top. book)

$\displaystyle \phi (t)=0$ for $\displaystyle t\leq 1/3$

$\displaystyle \phi (t)=1$ for $\displaystyle t \geq 2/3$

And so our homotopy could simply be

$\displaystyle F(x,t)=\phi (t) g(x) + (1-\phi (t))f(x)$

For that matter, we could simply chose

$\displaystyle F(x,t)=\sin (t) g(x) + \cos (t) f(x)$

But the problem with both these functions is that they don't nessicarily satisfy $\displaystyle \| F(x,t)\| = 1$, which is required if we want the range to be the n-sphere. So does anyone have any suggestions on how to impose that condition, or any other approaches to this problem that might prove fruitful?

Thanks.