Suppose that the power series $p(x)=\sum b_nx^n$ converges for $|x|\le 1$.Suppose that for some $\delta >0$, $p(x)=0$ for $|x|<\delta $.
Show that $b_n=0$ for all $n$.
My effort:The series $p(x)=\sum b_nx^n$ represents an analytic function and the power series has uncountably many zeros since $p(x)=0$ in $B(0,\delta)$.
Hence $p(x)=0$ which in turn $\implies b_n=0$
Is the solution okay ?