Suppose that $$\sum\limits_{n=1}^\infty a_{n}x^{n}$$
converges for $x \in (-R,R)$.
Show that if $f(x)=0$ for all $x \in (-R,R)$ then $a_n=0$ for all $n$.
When I look at this , my guess is that the answer involves showing that the radius of convergence is infinity?
i.e showing $R=\infty$. Is this a correct approach?
Assume that for some $N\geqslant 1$, we have $a_1=\dots=a_{N-1}=0$ and $a_N\neq 0$. Then write $$|a_Nx^N|=\left|\sum_{k=1}^Na_kx^k\right|\leqslant \left|\sum_{k=N+1}^\infty a_kx^k\right|,$$ which gives $$\tag{$\star$}|a_Nx^N|=|x^{N+1}|\left|\sum_{k=0}^\infty a_{N+1+k}\cdot x^k\right|.$$ Using the fact that the series $\sum_n a_nx^n$ converges on $(-R,R)$, we can show that the function $x\mapsto \sum_{k=0}^\infty a_{N+1+k}\cdot x^k$ is bounded on a neighborhood of $0$. Using $(\star)$ for $x\neq 0$, then letting $x\to 0$, we get a contradiction.