Let $f(x)=\sum\limits_{n=0}^\infty a_nx^n$ a power series and $f(0)\ne0$. (w.l.o.g. $f(0)=1$) Suppose the power series has radius of convergence $r>0$.
A power series is continuous in her convergence interval.
So there is a $\delta\in]0,r[$ so that for $|x|<\delta$ it's $|a_1x+a_2x^2+\dots|<1$.
My Questions:
why is $|a_1x+a_2x^2+\dots|<1$?
why did they choose $...<1$? Why $1$ ?
Thanks for helping!!