# Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend:

Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$.

The solution is apparently fairly simple, but I am having trouble getting a start on this problem. Any help?

• Note first that $\mathbb{R}\times\mathbb{S}^n$ smoothly embeds in $\mathbb{R}^{n+1}$ for each $n$, via $(t,\textbf{p})\mapsto e^t\textbf{p}$.
• Taking the Cartesian product with $\mathbb{R}^{m-1}$, we find that $\mathbb{R}^m\times\mathbb{S}^n$ smoothly embeds in $\mathbb{R}^{m}\times\mathbb{R}^n$ for each $m$ and $n$.
• By induction, it follows that $\mathbb{R}\times\prod_{i=1}^m \mathbb{S}^{n_i}$ smoothly embeds in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$.