Peter-Weyl theorem versions Let $G$ be a compact group. I learned the version of the Peter-Weyl theorem which says: the matrix coefficients of $G$ are dense in $L^2(G)$. Call this Peter-Weyl I. 
Apparently there is another version which states: For any $g \in G$ there exists a finite dimensional unitary representation $(\pi,V)$ such that $\pi(g) \neq I$ (identity). Call this Peter-Weyl II.
Can one prove Peter-Weyl II using Peter-Weyl I? A short slick proof is what I'm looking for, of course.

This answer is actually using a stronger version of Peter Weyl I, which state that the matrix coefficients are even dense in $C(G)$, the space of continuous functions. (So that might not answer your question)
First we show that $G$ must have a faithful representation. 
To show that, let $g\in G\setminus \{e\}$ and $f$ be continuous on $G$ so that $f(g) \neq  f(e)$. Then by Peter Weyl I, there is a matrix coefficient $u$ so that $u(g) \neq u(e)$. If this $u$ comes from the representation $\phi_1 : G \to GL(V)$, then we see that $g\notin  \ker(\phi)$. Let $K_1$ be this kernel. $K_1$ is a closed subgroup of $G$ which is strictly smaller then $G$. 
We are done if $K_1 = \{e\}$. If not, then we find $g_2\in K_1\setminus \{e\}$ and similarly construct a representation $\phi_2$ on $G$ so that the kernel $\tilde K_2$ has the property that $K_2:= \cap K_1\neq K_1$. 
Then inductively we find a descending sequence of closed subgroups 
$$G \supsetneqq K_1 \supsetneqq K_2 \supsetneqq K_3 \supsetneqq \cdots $$
But this chain must stability by dimension consideration and the fact that $G$ is compact. Thus $K_m = \{e\}$ for some $m$. Then the representation 
$$\Phi = \phi_1 \oplus \cdots \oplus \phi_m$$
is injective: if $g\in G$ and $\Phi(g) = 0$, then $g\in K_i$ for all $i=1, 2\cdots, m$. In particular $g = e$. 
Now the claimed results (Peter Weyl II) follows from the fact that every representation of a compact Lie group admits a $G$-invariant metric. You may check more in the book representation of compact Lie groups, where I learnt the above argument from this book. 
