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Let $(P_{nk})_{k \geq 1}$, $n=1,2,\cdots$ and $(P_{k})_{k \geq 1}$ be a sequence of nonnegative numbers satisfying $\sum_{k=1}^{\infty}P_{nk}=1$ and $\sum_{k=1}^{\infty}P_{k}=1$, and let $\lim_{n \to \infty}P_{nk}=P_{k}$, for each $k \geq 1$. Show that $\sum_{k=1}^{\infty}|P_{nk}-P_{k}|\to 0$.

I just found out that my solution is completely wrong. I attempted something using dominated convergence, but the limit I used was the limit of $p_{nk}-p_{k}$ instead of the limit if it's absolute value, so it's no good. Someone else tried to explain it to me using positive and negative parts of $P_{nk}$ and $P_{k}$, but I didn't really understand what they were trying to say.

Could someone please explain to me a detailed solution to this problem (pretend like you're trying to explain it to somebody who has no idea what they're doing; it might not actually be that far off at this point...)?

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Notice that the assumption $\sum_kp_k=1$ is necessary, otherwise take $p_{n,k}=1$ if $n=k$ and $0$ otherwise.

We have $$\sum_{k\geqslant K+1}p_{n,k}=\sum_{k\geqslant K+1}p_k+\sum_{k=1}^Kp_k-p_{n,k},$$ hence for each $\varepsilon\gt 0$ there is $K$ such that for each $n$, $\sum_{k\geqslant K+1}p_{n,k}\leqslant\varepsilon$.

This gives a necessary and sufficient condition for weak convergence for discrete random variables.

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