In stein's Fourier Analysis text i'm attempting to verifying my proof that the $\sum_{n=1}^{\infty}c_{n}$ converges to a finite limit $s$ then the series is Abel summable to $s$. My initial attempt can be seen in $(1.)$

$(0.)$

A series of complex numbers $\sum_{}^{}c_k$ is said to be Abel Summable to $s$ if for every $0 \leq r < 1$, the series in $(0.1)$ converges and the following limit occurs in $(0.2)$

$(0.1)$ $$ A(r)=\sum_{}^{}c_{k}r^{k}$$

$(0.2)$ $$\lim_{r \rightarrow 1}A(r) = s$$

$(1.)$

Assuming $s = 0$, then we have to show the following in $(1.0)$:

$(1.0)$ $$S_n = c_1 + \cdot \cdot \cdot + C_{N}$$, then: $$\sum_{}^{}c_{n}r^{n}=(1-r)\sum_{}^{}s_nr^{n}+s_{N}r^{N+1}.$$

$Lemma \, (2.0)$:

Applying Abel Summability as defined in $(0.1)-(0.2)$, the following conclusion is yielded:

$$A(r)=\sum_{}^{}C_{n}r^{n} = \lim_{r \rightarrow 1}(1-r)\sum_{}^{}S_nr^{n}=0$$.

From the previous result, it can be noted that on the RHS side of our series, can algebraically manipulated as follows: $$\lim_{r \rightarrow 1}(1-r)S_{0}r^{0} + S_{1}r^{1} + S_{2}r^{2} + \cdot \cdot \cdot \cdot + S_{n}r^{n}=0$$.

Since our series converges to zero for every $0 \leq r < 1$, concluding our proof.