Sequence $(a_n)_{n\in \mathbb{N}}$ such that $0 \leq a_n$ and $\sum_n a_n = 1$ . We also know that $a_n $ is non-zero for infinitely many n . For each $j \in \mathbb{N}$ we have non-negative sequence $(b_{j,n})_{n \in \mathbb{N}}$ such that $b_{j,n} \in [0,1] $ and $b_{j,n} \uparrow a_j $ as $n \rightarrow \infty$ .
Then prove that $\text{lim}_{n \rightarrow \infty} \displaystyle\sum_{j=1}^n jb_{j,n} = \displaystyle\sum_{j=1}^{\infty}ja_j$ ( possibly both are infinite )
Note : This result was used in probability class without proof , while computing expectations of discrete rv's