Exchange of limit and sum for a uniformly convergent sequence

Question

Let $\{x_n^k\}$ be a set of nonnegative numbers with $n$ being in the integers, and $k$ being natural numbers, such that the $x_n^k \rightarrow 0$ uniformly over $n$ as $k$ tends to infinity and $\sum_{n}x_n^k$ is finite for each $k$. Then is it justified to say $\lim_{k \to \infty} \sum_{n}x_n^k= \sum_n \lim_{k \to \infty}x_n^k=0$ ? I think my logic is right but want to check for sure

Answer 1

No, this is not true. Let $x_n^{k}=\frac 1 k$ for $|n| \leq k$ and $0$ for $|n| >k$. Then all the conditions are satisfied and $\sum_n x_n^{k}=\frac {2k+1} k \to 2$.