Question:
Let $\chi_1$ be the trivial character.
Let $\chi$ be a non-trivial character such that for all $g$, $\chi(g)$ is real, and $\chi(g)\geq 0$.
Show that $\langle\chi,\chi_1\rangle$ is strictly positive, and conclude that $\chi$ is reducible.
My partial solution:
$$\langle\chi, \chi_1\rangle = \frac{1}{|G|}\sum_{g\in G} \chi(g) \overline{\chi_1(g)}$$ $$= \frac{1}{|G|} \sum_{g \in G} \chi(g)$$ $\chi(1) \geq 1$ and $\chi(g) \geq 0 \quad \forall g \in G$ so $$\chi(g) \gt 0 \quad \forall g \in G$$ so $$\langle\chi, \chi_1\rangle \gt 0$$ How do I deduce that $\chi$ is reducible?