We know each character on dual group of $Z; \widehat{Z}$, is positive definite and if $\chi \in \widehat{Z}$ then $\left\| \chi \right\|_{\infty}=\chi(1)$. But I can not prove the problem that:
Let $\chi \in\widehat{Z}$ and $(\chi_{n})\subset\widehat{Z}$. Is when $\chi_{n}(1)\rightarrow \chi(1)$, then $\Vert\chi_{n}-\chi\Vert_{\infty}\rightarrow 0$ ?