This is an old qual problem I'm working on: Let $\{a_k\}$ and $\{b_k\}$ be two infinite sequences of real numbers. Suppose that $a_k>0$ and $0\leq b_k \leq 1$ for all $k\geq 1$, and suppose that $\sum_{k=1}^{\infty} a_k= +\infty$. Prove that there is an increasing sequence $\{k_n\}$ of positive integers such that
i) $\sum_{n=1}^{\infty} a_{k_n}= +\infty$
ii) $\lim_{n\rightarrow \infty} b_{k_n}$ exists.
I tried to show that $\{a_n\}$ has a divergent subsequence whose all subsequences are again divergent. Then, clearly this will finish the problem, because any subsequence of ${b_n}$ has a convergent subsequence, by the compactness of $[0,1]$. However, what I tried to show might be very false too, I'm not sure. I would appreciate any help. Thanks!