I want to prove that every power series is continuous but I am stuck at one point.
Let $\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ a power series with a radius of convergence $r>0$ and let $D:=\{x\in\mathbb R:|x-x_0|
Proof: Let $x\in D$ and $r_0\in\mathbb R^+$ such that $|x-y_0|
Since the power series converges uniformly on the closed disk $|y-x_0|\leq r_0$, we may choose a positive integer $N_\varepsilon$ such that $|\phi_{N_\varepsilon}|<\frac\varepsilon3$ for all $|y-x_0|\leq r_0$.
Since $S_{N_\varepsilon}$ is polynomial we can choose a $\bar\delta>0$ such that $|S_{N_\varepsilon}(y)-S_{N_\varepsilon}(x)|<\frac\varepsilon3$. So we get $|S(y)-S(x)|<\frac\varepsilon3+\frac\varepsilon3+\frac\varepsilon3=\varepsilon$ for $|y-x|<\delta$
Why is in the violet term $|\phi_{N_\varepsilon}|<\frac\varepsilon3$ correct? Don't I have two terms in the absolute value? Thanks for helping!