Let $G$ be a finite group and let $\rho : G \to GL(V)$ be an injective representation. I need to prove that each irreducible representation of $G$ is contained in $\otimes_{i=1}^{n} \rho$ for some $n \in \mathbb{N}$ (the tensor product of two representations $\rho : G \to GL(V)$ and $\varphi : G \to GL(W)$ is given by $\rho\otimes\varphi : G \ni g \mapsto \rho(g)\otimes \varphi(g) \in GL(V \otimes W)$).
I don't know how to do this, in particular, I don't see how the injectivity hypothesis can be applied. Any help would be greatly appreciated.
We have $\chi_{\rho^{\otimes n}} = \chi_\rho^n$. So, if $\varphi$ is another representation, $\langle \varphi, \chi_{\rho^{\otimes n}} \rangle = \frac{1}{|G|} \sum_g \chi_\varphi(g) (\overline{\chi_\rho (g)})^n$. We can write this as $\sum_b a_b b^n$ for some distinct $b$ and constants $a_b$.
Since the functions $\{b^n\}$ are linearly independent, if we had $\langle \varphi, \chi_{\rho^{\otimes n}} \rangle=0$ for all $n$, we would have $a_b=0$ for all $b$. In particular, $a_d=0$, where $d=\dim \rho$. But $\rho$ is injective, so $\chi_\rho(g)=d \iff g=1$, and we have $a_d = \frac{1}{|G|} \chi_\varphi(1)$. Therefore $\dim \varphi = \chi_\varphi (1)=0$, so $\varphi = 0$.