sum of sequences converging to $0$

Question

Let $I$ a countable set of indices, and for any fixed $i\in I$ let $(a^{(n)}_i)_{n\in\mathbb N}$ be a sequence with values in a non-archimedean local field (e.g. $\mathbb Q_p$). Assume that the following hypotheses hold:

1. For any $i\in I$, $\lim_{n\to\infty }a^{(n)}_i=0$.
2. For any fixed $n$, all but finitely $a^{(n)}_i$ are equal to $0$. In other words the sum $\sum_{i\in I }a^{(n)}_i$ is finite.

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Is it true that:

$$\lim_{n\to\infty} \sum_{i\in I }a^{(n)}_i=\sum_{i\in I }\lim_{n\to\infty}a^{(n)}_i=0 \;\;\;\;\;\text{?}$$

Remember that for non-archimedean fields: $\sum_na_n$ converges if and only if $a_n\to 0$

Answer 1

Consider a list of sequences:

$$ a_1^{(n)}=1,0,0,0,0,\dots $$ $$ a_2^{(n)}=0,1,0,0,0,\dots $$ $$ a_3^{(n)}=0,0,1,0,0,\dots $$ $$ a_4^{(n)}=0,0,0,1,0,\dots $$ $$ \vdots $$

Then $\sum_{i\in I }a^{(n)}_i=1$ for any $n$.