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Let $G$ be a compact Lie group. I have to prove that for $n$ large I there is an immersion of $G$ in the unitary group $U(n)$. I know that any finite-dimensional representation of a compact Lie group is unitary and thus completely reducible. So I have to prove that this representation is faithful... right?

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You need to prove that you group $G$ has a faithful representation. This "is" Peter-Weyl theorem (or a corollary of, depending on how one states things).

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