Positive-degree norm-one virtual characters are irreducible ($⟨\psi ,\psi ⟩=1,\psi (1)>0⟹\psi \in \mathrm{Irr}(G)$)

Write the virtual character as $\psi ={\sum }_{\chi }{a}_{\chi }\chi$ with integer coefficients and irreducible characters $\chi$. Character orthogonality gives $⟨\psi ,\psi ⟩={\sum }_{\chi }{a}_{\chi }^2$. Norm one forces exactly one coefficient to be $1$ or $-1$, and positive degree selects the positive sign.