 Euclidean embedding by finite cell attachment

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.