Characterize those power series that converge uniformly on ($-\infty$, $\infty$).
I can't understand why he wrote, "Now this is only possible if an+1=0." for any epsilon, there is another N and another an+1 it seems he only proves that an+1 converges to 0 but not equal to 0. could someone assist?
Consider the function $$\begin{array}{rcl} f:\Bbb R &\longrightarrow& \Bbb R\\ x & \mapsto & \vert a(x-c)^n\vert \end{array}$$ for some constants $a,c\in \Bbb R$ and some $n\in \Bbb N_0$.
If $a,n \neq 0$ we find that $f(x_n)\rightarrow \infty$ for $x_n \rightarrow \pm \infty$. So if $f$ is bounded ie. satisfies $\forall x: f(x) < C$ for some $C\in \Bbb R$, then either $a$ or $n$ has to be 0. If we know that $n\geq 1$, $a$ has to be 0.
