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

Lemma
On $\widehat G_\infty := \ds \coprod_{n \mathop \in \N} G_n$ the relation:
 * $\tuple {x_n, n} \sim \tuple {y_m, m} \iff \exists k \ge n, m: \map {g_{n, k} } {x_n} = \map {g_{m, k} } {y_m}$

is an equivalence relation.

Reflexivity
Since $g_{n, n} = \mathop {Id}_{G_n}$ we have:
 * $\forall \tuple {x_n, n} \in \widehat G_\infty: \map {g_{n, n} } {x_n} = \map {g_{n, n} } {x_n}$

Hence:
 * $\tuple {x_n, n} \sim \tuple {x_n, n}$

Symmetry
Let $\tuple {x_n, n} \sim \tuple {y_m, m}$.

Then there exists a $k \ge n, m$ such that:
 * $\map {g_{n, k} } {x_n} = \map {g_{m, k} } {x_m}$

Hence also:
 * $\map {g_{m, k} } {x_m} = \map {g_{n, k} } {x_n}$

That is:
 * $\tuple {y_m, m} \sim \tuple {x_n, n}$

Transitivity
Let $\tuple {x_n, n} \sim \tuple {y_m, m}$ and $\tuple {y_m, m} \sim \tuple {z_r, r}$.

Then there exist $k \ge m, n$ and $l \ge n, r$ such that:
 * $\map {g_{n, k} } {x_n} = \map {g_{m, k} } {y_m}$
 * $\map {g_{m, l} } {y_m} = \map {g_{r, l} } {z_r}$

Let $q:= \max \set {k, l}$.

Then we have:

that is:
 * $\tuple {x_n, n} \sim \tuple {z_r, r}$