How did Rudin conclude his argument there is no "boundary" between convergent and divergent series?

I lost my baby Rudin book on real analysis book but I recall a pair of results in homework exercises that he seemed to indicate that there is no "boundary" between convergent and divergent series of positive decreasing terms. One result was that if $a_n$ is positive decreasing, and $\sum_n a_n$ is divergent, then $\sum_n a_n/s_n$ is also divergent where $s_n$ is the $n$th partial sum of the $a_i$. So, since the original series diverges, we can keep repeating this construction to get series that diverge slower and slower, since $\lim_{n \to \infty} s_n = \infty$.

Rudin paired this with another homework exercise result about convergent series, something showing that given any convergent series (possibly of positive decreasing terms) you could construct a new series that converged "slower" in some obvious sense. Can someone recall that result?

Solution 1:

If $\sum a_n = \infty,$ then we can find $n_1 < n_2 < \dots $ such that $\sum_{n_k\le n < n_{k+1}} a_n > 1$ for all $k.$ Define $b_n = a_n/k$ for $n_k\le n < n_{k+1}.$ Then $\sum b_n =\infty,$ and $b_n/a_n \to 0.$

If $\sum a_n < \infty,$ then we can find $n_1 < n_2< \dots $ such that $\sum_{n_k\le n < n_{k+1}} a_n < 1/2^k$ for all $k.$ Define $b_n = ka_n$ for $n_k\le n < n_{k+1}.$ Then $\sum b_n <\infty,$ and $b_n/a_n \to \infty.$