Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I hope what I understood was really correct, any suggestion, comment or advice it would be great, thanks so much):
Let $X$ be a finite set, let $m$ be an integer, and for each $x\in X$, let $(\,a_n(x)\,)_{n=m}^{\infty}$ be a convergent sequence of real numbers. Show that the sequence $\big (\sum_{x\in X} a_n(x)\big)_{n=m}^{\infty}$ is convergent and $\lim_{n \rightarrow \infty}\big (\sum_{x\in X} a_n(x)\big)= \sum_{x\in X} \lim _{n \rightarrow \infty}a_n(x)$.
Scratch work: We will use induction on the size of the set X. Let $P(n)$ be the statement: "For any set of cardinality $n$ [where $n\in \mathbb{N}$] the above assertion is true, i.e., the sequence $(\,a_n(x)\,)_{n=m}^{\infty}$ is convergent an its limit is $\sum_{x\in X} \lim _{n \rightarrow \infty}a_n(x)$".
The base case is when $n=0$, then the set $X$ is empty. And since $\sum_{x\in \emptyset} a_n(x)=0$ as we have shown. Thus, $\lim_{n \rightarrow \infty}\big (0)=0$ and clearly the zero sequence is convergent, also we have $\sum_{x\in \emptyset} \lim _{n \rightarrow \infty}a_n(x)=0$ which prove the base case.
Now suppose we have proved the assertion for any set of cardinality $n\ge 0$. We may assume $\#X=n+1$ and we will show that the claim hold. Notice that the set cannot be empty because its cardinality. So at least has one element lies in it, let call it $x_0$. Let define $X' = X-\{x_0\}$.
Thus,
$\lim_{n \rightarrow \infty}\big (\sum_{x\in X} a_n(x)\big)= \lim_{n \rightarrow \infty}\big (\sum_{x\in X' \cup \{x_0\}} a_n(x)\big)=\lim_{n \rightarrow \infty}\big (\sum_{x\in X'} a_n(x)+a_n(x_0)\big)$; which is possible, since $X'\cap \{x_0\}=\emptyset$. Using the limit laws we can conclude the following:
$\lim_{n \rightarrow \infty}\big (\sum_{x\in X'} a_n(x)\big)+\lim_{n\rightarrow\infty}\,a_n(x_0)$. Since $\#X'=n$, we can use the inductive hypothesis and thus we have: $\sum_{x\in X'}\lim_{n \rightarrow \infty} a_n(x)+\lim_{n\rightarrow\infty}\,a_n(x_0)$. We know both are convergent, then their sum converges. Furthermore, $\sum_{x\in X'}\lim_{n \rightarrow \infty} a_n(x)+\sum_{x\in \{x_0\}}\lim_{n\rightarrow\infty}\,a_n(x)$ and this is equal to $\sum_{x\in X} \lim_{n \rightarrow \infty}a_n(x)$ which close the induction and give us the proof of the statement.
Thanks in advance.