Existence and Uniqueness of Direct Limit of Sequence of Groups/Lemma 1

Lemma
On $\hat{G}_{\infty} := \coprod_{n \in \mathbb{N}} G_n$ the relation
 * \[\left({x_n, n}\right) \sim \left({y_m, m}\right) :\iff \exists k \geqslant n,m: g_{n,k} \left({x_n}\right) = g_{m,k} \left({y_m}\right)\]

is indeed an equivalence relation.

Proof
Since $g_{n,n} = \mathop{Id}_{G_n}$ we have for all $(x_n,n) \in \widehat{G}_{\infty}$, $g_{n,n}(x_n) = g_{n,n}(x_n)$, hence \[			(x_n,n) \sim (x_n,n). \]
 * Reflexivity

Let $(x_n,n) \sim (y_m,m)$, then there exists a $k \geqslant n,m$ such that $g_{n,k}(x_n) = g_{m,k}(y_m)$ hence also $ g_{m,k}(y_m) = g_{n,k}(x_n)$, i.e. $(y_m,m) \sim (x_n,n)$.
 * Symmetry

Let $(x_n,n) \sim (y_m,m)$ and  $(y_m,m) \sim (z_r,r)$. Then there exist $k \geqslant m,n$ and $l \geqslant n,r$ such that \[			g_{n,k}(x_n) = g_{m,k}(y_m), \quad g_{m,l}(y_m) = g_{r,l}(z_r). \]		Let $q:= \max\{k,l\}$. Then we have \[			g_{n,q}(x_n) = g_{k,q}(g_{m,k}(y_m)) = g_{m,q}(y_m) = g_{l,q}(g_{m,l}(y_m)) = g_{l,q}(g_{r,l}(z_r)) = g_{r,q}(z_r), \]		i.e. $(x_n,n) \sim (z_r,r)$.
 * Transitivity