Prove that either $r \geq 1$ or $r = \infty.$ Let $\{a_n \}_{n=0}^{\infty}$ be a sequence of real numbers such that $\sum\limits_{n=0}^{\infty} |a_n|^2 &lt; \infty.$ Consider the power series $\sum\limits_{n=0}^{\infty} a_n x^n$ with radius of convergence $r.$ Show that either $r \geq 1$ or $r = \infty.$
  


How do I prove that? Any suggestion or hint regarding this will be highly appreciated. 

Thank you very much for your valuable time.
You need to prove that
$$\sum_{n=0}^\infty a_nz^n$$
converges absolutely when $|z|&lt;1$. I suggest using Cauchy-Schwarz. By hypothesis,
the $\ell^2$-norm of the sequence $(a_n)$ is finite.
For any bounded sequence $(a_n)$ the power series $\sum a_n x^{n}$ converges for $|x| &lt; 1$ (by comparison with a geometric series). Hence the radius of convergence is  a real number $\geq 1$ or $+\infty$. Note that $\sum |a_n|^{2} &lt;\infty$ implies that $a_n \to 0$ and hence $(a_n)$ is bounded.