Dushnik-Miller Proof

Dushnik-Miller Proof

I am working through the Dushnik-Miller Theorem in Jech's Set Theory and
have a question to one part of the proof. For the one's who don't have the
book at their fingertips here Theorem 9.7:

For every infinite cardinal $\kappa$ holds $\kappa\to (\kappa, \omega)^2$.

For the proof he choses $\{A,B\}$ to be the partition of $[\kappa]^2$ and
for every $x\in\kappa$ $B_x := \{y\in\kappa:x<y \textrm{ and } \{x,y\}\in
B \}$.

He differs two cases.
The first is: for all $X\subseteq \kappa$ with cardinality $\kappa$ exists
$x\in X$ such that $|B_x \cap X|= \kappa$
The proof of this part is clear to me.

The second is: There exists a $S\subseteq \kappa$ with cardinality
$\kappa$ such that $|B_x \cap S|<\kappa$ for all $x\in S$.
In this part he differs between $\kappa$ regular and singular.
Here is my question. I don't really see why we do this distinction...

I hope someone can explain me with some more details this part of the proof!

Thank you, and all the best

Luca!