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We know that any closed manifold $X$ can be embedded into some Euclidian space $\mathbb{R}^n$ for sufficiently large $n\in \mathbb{N}$. What is the easiest way to see this fact? I have seen several proofs but I keep forgetting them.

My motivation to ask this is the fact that any vector bundle $E\rightarrow X$ can be embedded in some trivial bundle $\mathbb{R}^n\times X\rightarrow X$. Any embedding $X\subset \mathbb{R}^n$ is sufficient to prove this.

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What definition of a manifold are you working with? –  Bitwise Oct 3 '12 at 1:39
Smooth ones will do. Thank you for clarifying the question. –  M. K. Oct 3 '12 at 2:15

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