Pretty sure this is normal. The car just things your in it, and you have the keys. OR you not waiting for the alarm to arm.
A better test would be to put your windows down get out and lock the car. Wait a little bit and reach in and open the door. Your alarm should go off then!
"What if there is no tomorrow....There wasn't one today!!!" Bill Murray
'04 OB/GL Prem.Pkg./Sport.Pkg/Splash