Where can I find the Inscribed Rectangle Problem proof?

Actually,I think this video gives a proof of the inscribed rectangle problem. The last step, which said that it's impossible to embed a Möbius Strip in the upper half space with boundary on the curve in the equatorial plan is justified by the following :

If we had one, we could paste the boundary of a disk (the disk is embedded in the lower half space) on the curve and then obtain and embedding of $\Bbb{RP}^2$ (projective plane) in $\Bbb{R}^3$.

But it's impossible : By Alexander duality if $X$ embeds in $\Bbb{R}^3$, then $H_1(X)$ has no torsion (Hatcher, Corollary 3.45).

Does it sound good ? Sorry for my English.