A question about bounding character ratios

The following question arose in a research project, and I'm sure it must be well known. I even know a very indirect proof, and would love to know if anybody knows a simple one.

Here is the question. Let $G$ be a finite group and let $\rho$ be an irreducible representation of $G$, with dimension $d_\rho$. Let $\chi$ be the character of the representation. Then the problem is to show that $$|\chi(g)| \le d_\rho$$ for all element $g \in G$.

Any help or reference greatly appreciated! Thanks, Nathanael

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$\chi(g)$ is the sum of the eigenvalues of $\rho(g)$. These are roots of unity, so have absolute value one, and there are $d_\rho$ of them. Thus the triangle inequality gives the inequality you want.