Let me correct the misconception appearing in Greg's [answer](https://mathoverflow.net/a/8799), which also made its way into the nLab article [real analytic space](https://ncatlab.org/nlab/show/real+analytic+space):

>"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 

<cite authors="Whitney, H.">_Whitney, H._, [**Differentiable manifolds**](https://doi.org/10.2307/1968482), Annals of Mathematics  (2) 37 (1936), no. 3, pp. 645–680, [JFM 62.1454.01](https://zbmath.org/?q=an%3A62.1454.01), [MR1503303](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1503303), [Zbl 0015.32001](https://zbmath.org/?q=an%3A0015.32001).</cite>

it is 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:

> Many portions of the proofs are based on the Weierstrass approximation theorem, if the manifolds are closed; if they are open, this theorem must be replaced by a corresponding theorem on functions defined in open sets.  This and other theorems which will be useful may be found in a previous paper.

(the previous paper being [Whitney - Analytic approximation of functions defined in closed sets](https://doi.org/10.2307/1989708)).



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. 


  [1]: https://i.sstatic.net/o344v.png