Intuitive proof of Tucker's lemma or Borsuk-Ulam theorem I'm looking for a intuitive proof of Tucker's lemma and/or the Borsuk-Ulam theorem. The proof should not make use of topology, cohomology etc. as it should be understandable by undergraduates. 
Thanks in advance!

This is my intuition for the special case of $n=2$ (it cannot be easilly generalized to higher dimensions). If $f: S^2\to \mathbb{R}^2$ is antipodal and nowhere zero, than $g:=f/|f|$ is an antipodal map from $S^2$ to the circle $S^1$. Take two antipodal points in the equator, $A$ and $B$. Their images $f(A)$ and $f(B)$ are antipodal on the circle and the image of the half-circle $AB$ on the equator is mapped to some curve on the circle that winds around the circle $n+1/2$ times for some $n$. So, the image of the whole equator ($(AB)$+the antipodal halfcircle $(BA)$) winds around the circle $2n+1$ times, a nonzero number. The equator can be slipped towards the north pole and contracted in $S^2$. However, for its image, you cannot contract a curve in the circle that has a nontrivial winding number.
