2

I am struggling trying to understand the final part of the proof of Theorem 3.54 on Baby Rudin. Here's the Theorem

Let $\sum a_n$ be a conditionally convergent series. Suppose : $$ -\infty \leq \alpha\leq \beta \leq +\infty$$ Then there exist a rearrangement $\sum a'_n$ with partial sums $\{s'_n\}$ such that

$$ \lim\limits_{n\to\infty} \inf s'_n=\alpha \quad \quad \lim\limits_{n\to\infty}\sup s'_n=\beta$$

The complete proof can be seen here, in this question Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof. I didn't quite understand the last part, namely

Finally, it is clear that no number less than $\alpha$ or greater than $\beta$ can be a subsequential limit of the partial sums of (25).

How do i prove this statement exactly?

Nikolaj Di Rondò
  • 305

1 Answers1

6

Because the series is convergent $a_n\to 0$, thus if $\epsilon >0$ then there is an $N$ such that for all $n\geq N$, $$|a_n|<\epsilon$$ this implies that For all $n\geq N$, $$\alpha-\epsilon<s_n<\beta+\epsilon$$ This is the construction, once one gets bigger that $\beta$ or less that $\alpha$, one turns around and goes the other way.

Rene Schipperus
  • 40,478