By source of ammonia, I mean are you adding pure ammonia, using a raw shrimp/fish food, or doing a fish-in with a betta or other fish?
Usually when people do a fishless cycle they want their tank to be able to process 2-4ppm in 24 hours. That way when they add their fish, there be any spikes in ammonia or nitrite because there is more than enough bacteria to cope with the bioload.
If you don't have any source of ammonia being added to your tank, it's not going to cycle. Either your tapwater has ammonia in it, or you have had or have something in there that's causing the reading.
Also, do you have a heater in your tank? The bacteria we want grows best at high temperatures. I'm assuming since it's a betta tank you would have one, but if the water is cold, it's going to take longer to cycle.
Finally, what test kit are you using? Liquid or strips? There is a big discrepancy in the accuracy between them. It's always recommended that you choose liquid reagent test kits as they tend to be much more accurate once calibrated.