I have the following proof for the Cauchy condensation test in my lecture notes. I can understand all that has been carried out except for the conclusion. Can you help explain why the sequences are either both bounded or both unbounded?
By theorem 3.8, it's referring to the following:
A series of non-negative terms converges iff the sequence of partial sums is bounded above.
I personally find the wording of the last paragraph confusing, so maybe it'd help to see it that way:
Assume $(T_k)_k$ is convergent, and let $T\stackrel{\rm def}{=}\lim_{k\to\infty} T_k\in\mathbb{R}$. Then, for any fixed $n$, taking the limit as $k\to \infty$ in the first inequality (which holds, since $k\to\infty$ we have $n \leq 2^k-1$ for $k$ large enough) $$ S_n \leq \lim_{k\to\infty} T_k = T $$ and $(S_n)_n$ is a bounded series of non-negative terms: it converges by Theorem 3.8.
Assume $(S_n)_n$ is convergent, and let $S\stackrel{\rm def}{=}\lim_{n\to\infty} S_n \in\mathbb{R}$. Then, for any fixed $k$, taking the limit as $n\to \infty$ in the second inequality (which holds, since $n\to\infty$ we have $n \geq 2^{k-1}$ for $n$ large enough) $$ T_k \leq \lim_{n\to\infty} 2S_n = 2S $$ and $(T_k)_k$ is a bounded series of non-negative terms: it converges by Theorem 3.8.
This shows the equivalence.