The fact of Saul's falling on his own spear (better described as an "assisted suicide) may be deduced by the report of the Amalekite youth in the first chapter of 2 Samuel, which was originally not a separate book. The young man found Saul "leaning on his spear" and put the king out of his misery, as Saul had request his armor bearer do in the previous chapter. 

> David said
> to the youth who was reporting to him, “How do you know that Saul and
> his son Jonathan are dead?” 6 The youth reporting to him replied: “I
> happened to find myself on Mount Gilboa and saw Saul leaning on his
> spear, with chariots and horsemen closing in on him... 9 Then he
> said to me, ‘Stand over me, please, and put me to death, for I am in
> great suffering, but still alive.’ 10 So I stood over him and put him
> to death, for I knew that he could not survive his wound. I removed
> the crown from his head and the armlet from his arm and brought them
> here to my lord.”

**So the author of 1 Samuel probably knew of Saul's death from this report, which was passed down to him.**