Let me correct the misconception appearing in Greg's answer, which also made its way into this nLab article:
"Anyway, the result is much harder than what Whitney did, which is a very good but expected application of the Weierstrass approximation theorem."
What one can do with the Weierstrass approximation theorem alone, in combination with Whitney's smooth embedding theorem, is to prove:
If $M$ is a compact stably parallelizable manifold then it can be embedded in some ${\mathbb R}^N$ as a smooth algebraic subvariety.
If one looks at Whitney's proof (incidentally, I am unaware of any textbook treatment of this proof) in
Whitney, H., Differentiable manifolds., Ann. Math., Princeton, (2) 37, 645-680 (1936). ZBL62.1454.01.
it clear that the proof is not an application of Weierstrass' theorem but needs much more delicate approximation arguments which are based on Whitney's earlier work. Whitney is quite clear on this point in the introduction to his paper:
That said, Whitney indeed proves that every smooth $m$-dimensional manifold admits an embedding in ${\mathbb R}^{2m+1}$ such that the image is a real-analytic submanifold.