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Suppose a $d$ dimensional manifold $M$ is embedded in $\mathbb{R}^N$ with $N$ larger than $2d+1$. Look at $M$ from a random direction and notice that you can see all of it. Hence you can embed it in $\mathbb{R}^{N-1}$.
Formally one needs Sard's theorem, the map from $M\times M$ minus the diagonal to the unit sphere defined by $(x,y) \mapsto (x-y)/|x-y|$ can't be surjective.