I am struggling with a guided exercise in which I need to prove the inclusion of the n-th skeleton in a CW-complex is an n-equivalence (meaning the induced maps on homotopy groups are isomorphisms for all dimensions smaller then n, and surjective in dimension n).
this is to be done without using the cellular approximation theorem (the aim is to use this fact in proving the theorem).
the first step should be showing that given a map from the k-sphere, for k≤n, to the n-th skeleton plus some n+1 dimensional cell is homotopic to a map that factors through the n-th skeleton. this is easy in the case that the map does not fill the n+1 dimensional cell(just use linear homotopy away from the missing point) but i am not sure what to do in order to remove possibility of a "space filling curve" type situation.
I should add that using simplicial approximation is allowed