Group homomorphism and equivalence relation

Question

For each natural number $n$ we are given a group $G_n$ and a group homomorphism $\phi:G_n \rightarrow G_{n+1}$. Define a binary relation on the disjoint union $\amalg_{n=1}^{\infty} G_n$ of these groups. Given $x_n \in G_n $ , $x_m \in G_m$ , we define

$x_n \sim x_m \iff \exists k \geqslant max(m,n)$ such that $\phi_{k-1}...\phi_m(x_m)=\phi_{k-1}...\phi_n(x_n).$

1. How can I prove this is an equivalence relation?
2. Furthermore, prove that the quotient set $G_\infty=\amalg_{n=1}^{\infty}G_n/\sim$ is a group under the operation $[x_n] \cdot[y_n]=[{x_ny_n}]$ defined on elements $x_n,y_n \in G_n$ for all $n$.

I am having trouble setting the scene with the proof. If I am not wrong, the group $G_\infty$ is the inverse limit of the groups $G_n$.

Answer 1

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