A stuck in Theorem 8.2 of Baby Rudin’s PMA

analysiscontinuityconvergence-divergencelimitsreal-analysis

The following is Rudin's Theorem $8.2$ and its proof.
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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.

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Best Answer

Your proof has two issues:

  • In your application of 7.10, you've assumed $\sum c_n$ converges absolutely.
  • In your final steps, you've only written that $f$ is continuous on $(-1,1)$, which in no way justifies the evaluation of the limit as $x \to 1$.

Of these, the former is devastating and the latter is mostly cosmetic.

That is, dealing with them in reverse:

  • Your applications of 7.10 and 7.12 could have just used $E = [-1,1]$ instead of $E = (-1,1)$. With this change, you would be able to conclude from $f$ being continuous on $[-1,1]$ that the limit $\lim_{x\to 1^-} f(x) = f(1)$, which would be exactly what you wanted to show, if not for the fact...
  • Theorem 8.2 only assumes $\sum c_n$ converges, not that it converges absolutely. With this, your initial application of 7.10 is invalid.

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.

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