I was asked to prove the following statement:
$\mathbb{Z}\times\mathbb{Z} \not\subseteq \mathbb{Z}\times\mathbb{N}$.
Intuitively seems simple enough, but I have not been trained in rigorously proving inclusion cases with cartesian products.
Your help will be greatly appreciated and thank you beforehand for you welcome assistance.
This is as much as I have been able to do:
[![enter image description here](https://i.stack.imgur.com/WND2g.png)](https://i.stack.imgur.com/WND2g.png)