Prove that any two maps $S^m \to S^n$, where $m < n$, are homotopic.
I've been fiddling around with trying to use the Simplicial Approximation theorem since that's the material we've recently covered in class but still can't seem to be able to tackle this one.
Take cellular structures with each two cells (one point one cell of the dimension of the sphere, the points being chosen $x$ and the image of $x$ under the map of interest). Cellular approximation yields that any map $S^m \to S^n$ is homotopic to a map $f$ s.t. $f(S^m_i) \subset S^n_i$ (where $S^m_i$ denotes the $i$-skeleton), hence homotopic to a map $im(f)=f(S^m_m) \subset S^n_m = *$ i.e. the constant map.