Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$
This is my attempt so far.
Let us suppose for a contradiction that $ \exists g \in G$ such that for all non-trivial $\chi\in Irr(G)$ we have $\chi(g)=0.$
At this point I am now stuck. I assume that this has something to do with the orthogonality relations but I am unsure how to proceed.
Am I on the right path?