Now we claim $f_{N}(\theta)=\displaystyle\sum_{1\le|n|\le N}\frac{e^{in\theta}}{n}$ is uniformly bounded in $N$ and $\theta\in[-\pi,\pi]$
My attempt :
Note that $$f_{N}(\theta)=\sum_{1\le|n|\le N}\frac{e^{in\theta}}{n}=\sum_{n=-N}^{1}\frac{e^{in\theta}}{n}+\sum_{n=1}^{N}\frac{e^{in\theta}}{n}=\sum_{n=1}^{N}\frac{e^{in\theta}-e^{-in\theta}}{n}$$
Lemma : Suppose that the Abel means $\displaystyle\sum_{n=1}^{\infty} r^{n}c_{n}$ of the series $\displaystyle\sum_{n=1}^{\infty}c_{n}$ are bounded as $r\rightarrow 1$ for the left. If $c_{n}=O(\frac{1}{n})$ then the partial sum $\displaystyle\sum_{n=1}^{N}c_{n}$ is bounded .
So, in this case, we let $c_{n}=\displaystyle\frac{e^{in\theta}-e^{-in\theta}} {n}$ and of course $c_{n}=O(1/n)$ . So now we determined the series bounded or not as $r\rightarrow1^{-}$ .
For all $r\in[0,1)$ and $\theta\in[-\pi,\pi]$ , we have \begin{align} \sum_{n=1}^{\infty}r^{n}c_{n}&=\sum_{n=1}^{\infty}r^{n}\frac{e^{in\theta}-e^{-in\theta}}{n}\\ &=\sum_{n=1}^{\infty}r^{n}\frac{2i\sin(n\theta)}{n}\\ &=2i\sum_{n=1}^{\infty}r^{n}\frac{\sin(n\theta)}{n} \end{align}
Therefore, one has
$$2i\lim_{r\rightarrow 1^{-}}\sum_{n=1}^{\infty}r^{n}\frac{\sin(n\theta)}{n}=2i\sum_{n=1}^{\infty}\lim_{r\rightarrow 1^{-}}r^{n}\frac{\sin(n\theta)}{n}=2i\sum_{n=1}^{\infty}\frac{\sin(n\theta)}{n}\in {\bf C}$$
keep in mind that this series $\displaystyle\sum_{n=1}^{\infty}\frac{\sin(n\theta)}{n}$ is convergent for all $\theta\in[-\pi,\pi]$ by Dirichlet test , and the first equality that the limit and summation can switch since the series $\displaystyle\sum_{n=1}^{\infty}r^{n}\frac{\sin(n\theta)}{n}$ converges absolutely for all $r\in[0,1)$ and $\theta\in[-\pi ,\pi]$.
Whence , by lemma we have $$\sum_{n=1}^{N}c_{n}=\sum_{n=1}^{N}\frac{e^{in\theta}-e^{-in\theta}}{n}$$ is bounded for all $\theta\in[-\pi,\pi]$ and $N=1,2,...$
I feel some strange about my working , particularly , the tail of the prove , Can we conclude that the uniformness in $N$ , In fact that I am not very sure .
Can someone check my proof for validity if you have the time . Any comment or suggestion would be appreciated . Thanks for considering my request .
