Consider two sequences $\{a_n\}$ and $\{b_n\}$ with the two properties: (1) $a_n, b_n > 0$ for all $n$ and (2) $\sum_n a_n = \sum_n b_n = 1$. In addition,
$$
\frac{a_1}{b_1} > \frac{a_2}{b_2} > \cdots
$$
that is, $\frac{a_n}{b_n}$ is a decreasing sequence.
Let $A_n = \sum_{k=1}^n a_k$ and $B_n = \sum_{k=1}^n b_k$; the $n$-th partial sum of each sequence.
Then $$ \frac{A_n}{B_n} $$ is also decreasing?