The following is taken from Hungerford's undergraduate Abstract Algebra An Introduction text
Lemma 1: Let $M$ and $N$ be normal subgroup of a group $G$ such that $M\cap N=\langle e\rangle$. If $a\in M$ and $b\in M,$ then $ab=ba.$
(2) Let $I$ be the set of positive integers and assume that for each $i\in I, G_i$ is a group. The infinite direct product of the $G_i$ is denoted $\prod_{i\in I}G_i$ and consists of all sequences $(a_1,a_2,\ldots)$ with $a_i\in G_i.$ Prove that $\prod_{i\in I}G_i$ is a group under the coordinatewise operation $$(a_1,a_2,\ldots)(b_1,b_2,\ldots)=(a_1b_1,a_2a_2,\ldots)$$
Exercise 1: With the notation as in (1), let $\sum_{i\in I}G_i$ denote the subset of $\prod_{i\in I}G_i$ consisting of all sequences $(c_1,c_2,\ldots)$ such that there are at most a finite number of coordinates with $c_j\neq e_j,$ where $e_j$ is the identity element of $G_j.$ Prove that $\sum_{i\in I}G_i$ is a normal subgroup of $\prod_{i\in I}G_i.$ $\sum_{i\in I}G_i$ is called the infinite direct sum of the $G_i.$
$\textbf{Proof}$ for lemma 1 for infinite direct sum of groups:
With the notation as in (2) and Exercise 1, let $M=\sum_{i\in I}M_i$ and $N=\sum_{i\in I}N_i$ where both $\sum_{i\in I}M_i$ and $\sum_{i\in I}N_i$ are normal subgroups of $\prod_{i\in I}G_i$ such that $M\cap N=\langle e\rangle,$ $e=(e_i)_{i\in I},$ and each $e_i$ is an identity elements of $G_i.$ Let $m\in M$ and $n\in N,$ where $m=(m_i)_{i\in I}, n=(n_i)_{i\in I},$ and at most finitely many of the $m_j$ and the $n_j$ are not equal to $e_j$ which is the identity element of $M$ and $N$. As in the hypothesis, let $a\in M, b\in N.$ We note that $a=b$ iff $a_i=b_i,$ where $a_i\in (a_i)_{i\in I}, b_i\in (b_i)_{i\in I},$ for all $i\in I.$ Let $a^{-1}b^{-1}ab\in \prod_{i\in I}G_i.$ Since $M, N$ are both normal subgroups of $\prod_{i\in I}G_i$, then $a^{-1}b^{-1}a\in N, b^{-1}ab\in M.$ By closure, $a^{-1}(b^{-1}ab)\in M, (a^{-1}b^{-1}a)b\in N.$ But $M\cap N=\langle e\rangle,$ so $a^{-1}b^{-1}ab=\langle e\rangle$ gives $ab=ba$.
Question: I would like to know if my proof is correct, also, I posted the proof for Exercise 1 here
Thank you in advance