Isomorphism for Infinite Graphs.

Suppose that $G$ and $H$ are infinite graphs and that $G$ is isomorphic to a subgraph of $H$ and $H$ is isomorphic to a subgraph of $G$. Must $G$ and $H$ be isomorphic?

I've only just started on graph theory and this is one of the bonus questions on the assignment sheet. I don't see anything in my notes regarding infinite graphs and so I don't think this can quite classify as homework.

My intuition tells me that this might be a good definition for isomorphism for infinite graphs and hence Yes to the question but I don't quite seem to have any ideas on how to prove it.

Any insight or literature on this (because I can't seem to find much on this even on the internet!) will certainly be helpful.

I think this example works: Vertices are non-negative integers and you put edges only between two consecutive integers (this is basically the positive x-axis with the integers as vertices). This is your $G$.

Erase one edge in $G$. This is $H$.

Show that this is a counterexample...