I'm reading this proof of the Borsuk-Ulam theorem on Guillemin & Pollack's 'Differential Topology' book:
I'm trying to complete the step where it must be shown that regularity for $f/|f|$ is equivalent to the condition of $f$ being transversal to $l$. I've followed the hint, which points to this result:
...which I've already proved. But I don't see exactly how the conclusion of the exercise should be used here. I take it that in this case, we should view $x\mapsto\frac{f(x)}{\left|f(x)\right|}$ as the composite of functions $f$ and $x\mapsto\frac{x}{\left|x\right|}$ (the latter would be our $g$ according to the exercise) and that $l$ would then be $g^{-1}(W)$, but I'm not sure about what $W$ would be and I don't know how to continue right now. Am I on the right track?