Fishless cycling question

This is only my second cycle total, the first one I did with fish, because I did not know any better.
I guess it is less a question rather than needing reassurance on what I am doing:
I added Ammonia to the completely set up tank (decoration, plants, ...) on Sunday to a level of 5 ppm, and checked my water daily since.
Monday and Tuesday I squeezed the floss of my other tank into the new tank, trying to get some bacteria in there.
Yesterday I read 0 Nitrites and below 10ppm Nitrates (from tap water).
Today I read 1-2 ppm Nitrites and only about 0.25 ppm Ammonia. My Nitrates went up to about 40 ppm.
I remeasured everything and got the same results (I use drop tests).

Now here are my questions:
Is this normal it can be so fast? My fist tank took 4 weeks to cycle!
Is it correct that I should replenish Ammonia to 2-3 ppm?

Also, to anyone who has not tried fishless cycling yet, I can only recommend it. It is so much more easier and relaxing not having to worry about the fish you let swim in a toxic "soup" all day!

Thank you for your reply!
I don't think it finished cycling, but it definitely started. My previous experience was just that I did not see any nitrites for about 10 days, and now it has only been 4 days and I was wondering if this is normal, to see it started at all after only a few days.
I will add some more ammonia and then recheck tomorrow. Let's hope all will be going well.