I am trying to represent the following set as the countable union of sets
$A = \{(x,y)|x<y\}$
I know that $A = \bigcup \{\{x|x<a\} \times \{y|a<y\}\}$ where $a$ is a real number
My problem is that I need to represent set $A$ using the countable union of sets using the cartesian product (as shown above), and in particular I need variable $a$ to be a function of natural numbers not reals. For example, $a$ can be $1/n$ where $n$ is a natural number ( this 1/n is wrong but I am just adding it for clarification).
Is such representation using natural numbers possible?
You can let your $a$ range over $\mathbb Q$ instead of $\mathbb R$.
If $x$ and $y$ are real numbers such that $x<y$, there is always a rational between them, so $(x,y)$ will be in one of your sets.
Hopefully you already know that $\mathbb Q$ is countable.
However, what that gives you is not $A$ as a union of countable sets, but $A$ as a countable union of sets that that are themselves not countable.
For an actual union of countable sets, there has to be uncountably many of those sets. And for that I don't think we can get better than something silly like $$ A = \bigcup_{x\in\mathbb R, d\in(0,1]} \{ (x,x+d+n) \mid n\in\mathbb N \} $$