I'm trying to prove that the best moment tail bound is no worse than the best Chernoff bound. I have all pieces other than this little frustration here:
if $\forall i, c \leq \frac{a_i}{b_i}$, then it follows that $c \leq \frac{\sum_{i=1}^\infty a_i}{\sum_{i=1}^\infty b_i}$.
My intuition is that both summations in numerator and denominator act as averages for $a_i$'s and $b_i$'s respectively, and thus $\frac{\sum_{i=1}^\infty a_i}{\sum_{i=1}^\infty b_i}$ could be no smaller than $\min \{\frac{a_i}{b_i}\}$. There must be something really basic that I must have overlooked...so thanks for the heads up in advance!
Assuming all numbers are positive and that the series are convergent, you have $$cb_i \le a_i$$ for all $i$. Hence, summing all up, you have $$c\sum_i b_i \le \sum_i a_i$$ which is equivalent to $$c \le \frac{\sum_i a_i}{\sum_i b_i}$$