Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that

$\lim \sup_{k \rightarrow \infty} \frac{a_{k+1}}{ (\prod^k a_j)^{2 +2/\epsilon + \delta}} \frac{1}{b_{k+1}} = \infty$ and for every suf. large $k$ we have $\sqrt[1+\epsilon]{ \frac{a_{k+1}}{b_{k+1}} }\geq \sqrt[1+\epsilon]{ \frac{a_{k}}{b_{k}} }+1$. Why is the infinite series $\sum^\infty \frac{b_k}{a_k}$ summable? (This claim comes from a theorem in a published paper whose proof we are formalising using the proof assistant Isabelle/HOL. Summability easily follows if we strengthen the assumption substituting $\lim \sup$ with $\lim$, but is there a way to show summability without strengthening any assumption?) Thank you in advance!