Consider the following fact from algebraic geometry:

Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set.

I know how to prove this fact using Hironaka's and Nagata's theorems. However it looks much more elementary (at least more elementary then Hironaka's theorem). In particular, it is obvious for quasi-projective varieties.

So here are my questions:

Does this fact have an elementary proof?

Does anybody know the analogous fact for ($C^\infty$) Nash manifolds?

Thank you very much.