Suppose that $\{x_{i,n}:1\le i\le n,n\ge1\}$ is a real-valued triangular array such that the limit $\lim_{n\to\infty}\sum_{i=1}^nx_{i,n}$ exists and $x_{i,n}\to x_{i,\infty}$ for each $i\ge1$ as $n\to\infty$.
Is it true that $ \lim_{n\to\infty}\sum_{i=1}^nx_{i,n}=\sum_{i=1}^\infty x_{i,\infty}? $ If not, when can we say that it is true?
It is quite easy to construct an example where this is the case (for instance, $x_{i,n}=a_ia_n$, where $\sum_{i=1}^\infty a_i$ converges and $\lim_{n\to\infty} a_n$ exists). I do not think that this is always the case even though I am struggling to construct a counterexample. It seems that we might be able to use the dominated convergence theorem, but we actually have just one limit and I do not think that the dominated convergence theorem is applicable in this situation.
Any help is much appreciated!
This is wrong: just take $x_{i,n}=\frac{1}{n}$. Then all your sums are $1$, but $x_{i,n} \underset{n \to \infty}{\longrightarrow} 0$, and $\sum\limits_n 0 \neq 1$.