An explicit embedding of $S^m \times S^n$ into $\mathbb R^{m + n + 1}$.
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Your solution is absolutely correct. To use the standard stereographic projection $s : S^{m + n +1} \setminus \{ N \} \to \mathbb R^{m + n +1}$, where $N = (0,\ldots,0,1)$ is the north pole, I suggest to work with the embedding $$\phi : S^m \times S^n \to S^{m + n +1}, \phi(x,y) = \frac{1}{\sqrt 2} (x,y) .$$ Clearly $N \notin \phi(S^m \times S^n)$.