Rudin proves a statement in his classical book. Most of this proof is straightforward, while a subtle point confuses me.
Claim: Suppose $a_1 \geq a_2 \geq \dots \geq 0$. Then, the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the series $\sum_{k=0}^{\infty} 2^k a_{2^k}$ converges.
Proof: Let $s_n=a_1+a_2+\dots +a_n$, and $t_k = a_1 + 2a_2 + \dots +2^k a_{2^k}$. Obviously, if $n<2^k$, then $s_n \leq t_k$; if $n>2^k$, then $2s_n\geq t_k$.
Then, Rudin concludes that $s_n$ and $t_k$ are both bounded or both unbounded. Well, what if $n<2^k$ and $t_k$ is unbounded? In this case, we have no idea whether $s_n$ is bounded or not. Thanks for any explanation.