The following is Rudin's Theorem $8.2$ and its proof.
The proof seems a bit hard for readers like me. So, I have tried an approach to its proof using partial sums and applying Theorem $7.8$ & Theorem $7.12$ as follows.
Proof
Consider a sequence $\left\{s_n\right\}$ of partial sums $s_n(x)=\sum_{i=0}^{n}{c_i\ x^i}$, where $\left|x\right|<1$ and where $\sum {c_n}$ converges.
Since $\sum\ c_n$ converges, for every $\varepsilon>0$, there exists an integer $N$ such that $\sum_{i=m}^{n}\left|{c_i}\right|<\varepsilon$ if $m,n>N$.
Then,
$\left|s_n\left(x\right)-s_m\left(x\right)\right|=\left|\sum_{i=0}^{n}{c_ix^i}-\sum_{i=0}^{m}{c_ix^i}\right|$
$$= \left|\sum_{i=m+1}^{n}c_i x^i\right|\le\sum_{i=m+1}^{n}\left|c_i\right|\le\sum_{i=m}^{n}\left|c_i\right|<\varepsilon,\ \text{if}\ m,n>N.$$
Thus, by Theorem $7.8$, Cauchy-criterion, ${s_n}$ converges uniformly to $f$ given by $f(x)=\sum_{n=0}^{\infty}{c_n\ x^n\ }$.
Since $s_n$ is the sum of a finite number of continuous functions on $[-1,1]$, $s_n$ is continuous on $[-1,1]$, too.
Then, by Theorem $7.12$, the limit function $f$ is continuous on $[-1,1]$.
Then, we obtain
$$\lim_{x→1}\;f(x)=\lim_{x→1}∑_{n=0}^∞\;c_n x^n =∑_{n=0}^∞c_n .$$
\qed
I am wondering if my attempt is valid or not and is it isn't please suggest me. Thanks.
The following are the theorems I used in my attempt.
Best Answer
Your proof has two issues:
Of these, the former is devastating and the latter is mostly cosmetic.
That is, dealing with them in reverse:
The good news is that your proof can be immediately modified to prove the corollary of 8.2 which is identical to 8.2 except that we strengthen the assumption to $\sum c_n$ converges absolutely. The bad news is that this weakening of 8.2 is much less useful, and obviously it fails to provide an alternative proof of 8.2 itself.