Suppose I have
(i) $\alpha_n \geq 0$, $\beta_n \geq 0$ such that
(ii) $\sum_{n=1}^\infty \alpha_n = \sum_{n=1}^\infty \beta_n = 1$, and
(iii) $\sum_{n=1}^\infty n\alpha_n < \infty$. Intuitively, if
(iv) $\forall N \in \mathbb{N} :\sum_{n=1}^N \alpha_n \leq \sum_{n=1}^N \beta_n$
then I should also get $\sum_{n=1}^\infty n\beta_n < \infty$.
(Sorry, there was a typo in the original equation (iv): the direction of the inequality was reversed). Here is my proposed solution, but I feel queasy about some steps so please point out any mistakes.
$\sum_{n=1}^\infty n \alpha_n = \sum_{n=1}^\infty \sum_{m=n}^\infty \alpha_n = \sum_{n=1}^\infty \left(1 - \sum_{m=1}^{n-1} \alpha_n \right) $
$\geq \sum_{n=1}^\infty \left(1 - \sum_{m=1}^{n-1} \beta_n \right) = \sum_{n=1}^\infty \sum_{m=n}^\infty \beta_n = \sum_{n=1}^\infty n \beta_n $
