SCHWEITZER 1999 Problem 10

Let $M = F_1\times\cdots\times F_k$ be the product of $k$ smooth, closed surfaces (2-dimensional, $C^\infty$ , compact, connected, manifold without boundary), $s$ of which are non-orientable. Prove that $M$ can be embedded in $\mathbb{R}^{2k+s+1}$ .