Why does the prosecutor focus on Mersault's behavior at his mother's funeral?

The prosecutor is trying to prove that Mersault has committed premeditated murder in the death of the Arab man.

The prosecutor is showing that Mersault did not feel badly about his mother's death. He didn't grieve for her which is evidenced by his napping at her funeral. He wants to show that this absence of grief is a sign of moral depravity and that is a threat to society. The moral decay that Mersault shows by not being upset demonstrates his guilt and warrants the death penalty.