Let $f(x) = \sum^∞_{n=0}{a_n x^n}$ be a real power series for some sequence $(a_n)^∞_{n=0}$ of real coefficients. Suppose that for some $y ∈ R$, the series $f(y)$ is convergent. Show that there exists $K > 0$ such that $|a_n y^n| \leq K$ for all $n \in \mathbb{N}$.
Proof:
As $f(y) = \sum^∞_{n=0}{a_n y^n}$ converges, $a_n y^n \to 0$ as $n \to ∞$, so the sequence $(a_n y^n)^∞_{n=0}$ is bounded. Therefore, $\exists K >0$ such that $|a_n y^n| \leq K$ for all $n \in \mathbb{N}$.
Would you add anything to this proof to get full marks or is this sufficient?