Engine RPMs on 2007 Pontiac G5

I drive a 2007 Pontiac G5 with an automatic transmitions. It does, however, have a tachometer also and I have noticed that the engine seems to prefer to shift at somewhere 3500 RPMs and 4000 RPM. There have been times, however, where I have had to hit the accleerator petal quickly like when I have had to pull into traffic on an Interstate Highway and the RPMs have dramaticly jumped to 5000 to 6000 RPM.
While the tachometer gauge "red-lines" at 7000 to 8000 RPMs, I just want to make sure that I have not done any damage to my engine when I have been forced to make it jump to 5000 to 6000 RPMs.

Please let me know. Thanks.

Depending on several engine inputs and engine load, vehicle speed and throttle position determines when the trans will upshift and or downshift. This shouldn't hurt the engine at all though.