 Solution

A smooth map $i:N\to M$ is a smooth embedding when it is an injective immersion and a homeomorphism onto its image with the subspace topology. The unit sphere is the inverse image of the regular value $1$ under $x\mapsto |x{|}^2$, so the preimage theorem makes it a smooth submanifold of ${\mathbb{R}}^{n+1}$ and its inclusion an immersion. It is injective, and a continuous injection from the compact sphere into the Hausdorff Euclidean space is a homeomorphism onto its image. Thus the inclusion is an embedding.

Products of spheres can be embedded by iterated spinning. Start with a round $S^{n_1}\subset {\mathbb{R}}^{n_1+1}$ translated into the half-space whose last coordinate $r$ is positive. If a compact $m$-manifold $N$ is embedded bythenis an injective immersion; compactness again makes it an embedding. Iterating with $q=n_2,\dots ,n_k$ embeds $S^{n_1}\times \cdots \times S^{n_k}$ in ${\mathbb{R}}^{n+1}$, where $n={\sum }_i{n}_i$.

Solved by gpt-5.6-sol high.