OK, so I have had my 20 gal tank with fish setup for over a month now, and I haven't seen any signs of nitrites.

I have a QT tank 10 gal, setup for a while now, and I decided to run in through the fishless cycling process.

I added ammonia (Ace HW brand) to the tank on Monday. I checked it again Monday night to see the levels of ammonia, and hit it again.
after testing that, my ammonia levels were between 4 and 8ppm.

I have the temp range right now at approximately 78deg, all the cheap heater will do I think.

I just finished testing the water of the QT tank, and I'm seeing 0 ammonia, between 2.0 and 5.0 ppm nitrites, and 5ppm nitrates.
Should the cycle process happen so quickly?

I did put in some live plants on a piece of driftwood that I got when I purchased a used 75gal, established tank.
Is it possible that some bacteria lived on the DW and plants, and survived the trip? It was kept in an unheated bucket for 2 or 3 days, filled with tank water from the established tank.

Is the tank properly cycling, or am I seeing false reading?

I put more ammonia it a few minutes ago, and I'll test that tomorrow morning.