My idea was to find a composition of functions that turnt $\mathbb S^{n+1}\setminus\{N,S\}$ into $D^{n}\setminus\{0\}$ (the proyecton of hyperplane of the first $n$ components) and then into $\mathbb S^n$ (with the normalization of the "points looked as vectors"), and move on with the properties of a deformation retract. But I don't know how further I can go with this idea.
It also sounds much more complicated that what it seems, but what do I know? Could anyone please help me out?
P.S.: To me, $A\subset X$ is a deformation retract if there exists a function $r:X\to A$ such that $r\circ i = id_A$ and $i\circ r \simeq id_X$ ($i$ is the inclusion and "$\simeq$" is the homotopy between two maps).