Subset relation in a sequence of sets (Proof by induction)

Question

I am trying to prove this claim:

Let $G$ be a set of ordered pairs (it is supposed to denote the graph of a relation on a set). Consider the following sequence of sets: $G_1= G$ and $G_{n+1}=G_n\cup G_nG_n$. For each $i, j\in\mathbb{Z}^+$, if $i\le j$, then $G_i\subseteq G_j$.

I know that the proof is supposed to be done by induction but I am having a hard time with what I'm supposed to carry out induction on. Should I assume that $G_i ⊆ G_j$ as my induction hypothesis to show that $G_{i+1} ⊆ G_{j+1}$ (I'm not even sure if that's valid for induction)? Any help would be much appreciated.

Answer 1

Base case(s) for all $i\in\Bbb Z^+$ we clearly see $G_i\subseteq G_i$.

For the induction show for all $(i,j)\in\Bbb Z^{+2}:i\leq j$ that if $G_i\subseteq G_j$ then $G_i\subseteq G_{j+1}$

(Ps: easily done by demonstrating for all $j\in \Bbb Z^+$ that $G_j\subseteq G_{j+1}$)

---

You are proving induction on iterations of $j$. $$\begin{split}\forall i\in\Bbb Z^+~&~P(i,i)\\ \forall i\in\Bbb Z^+~\forall j\in\Bbb Z^+~&~(P(i,j)\to P(i,j+1))\\\hline\therefore\quad \forall i\in\Bbb Z^+~\forall j\in\Bbb Z^+~&~P(i,j)\end{split}$$