Divergence of sequences and associated rate

Question

I'm given with two sequences of positive reals $\{a_n\}_{n \in \mathbb{N}}$ and $\{b_n\}_{n \in \mathbb{N}}$ living in a compact set $\Omega$ so that $\lim_\limits{n \to \infty} a_n = 0$ while $\lim_\limits{n \to \infty} b_n$ may not even exist, and if it does, it is different from zero. Armed with these, I would like to show that: $$ \lim_\limits{n \to \infty} \sum_{k = 1}^n b_k \le \lim_\limits{n \to \infty} \sum_{k = 1}^n a_k $$ does not hold true. Any suggestion on the argument I should make, or some counterexample? I'm tempted to say that the LHS diverges "faster" than the RHS, but this is of course not formal (and possibly not even true).

Answer 1

LHS diverges to infinity and RHS can converge or diverge to infinity. Remeber that $k_n\to 0$ is a necessary condition (but not sufficient) for that $\sum k_n$ converges.

If $\sum a_n$ converges the inequaility is false. But if $\sum a_n=\infty$ I'd say that is is true. For example $$\sum_{k=1}^\infty 1=\sum_{k=1}^\infty\frac1n=\infty$$