Hi everybody! I bought a betta last year and kept it in a 0,5G tank (I know

). He (Gustave) got sick maybe 3 weeks ago... he stopped eating, move and always stayed on top of the water, on the surface.
So I changed his food and bought BettaFix (form hints form a Pet Store). Gustave went a little better, at least he ate (or try to...) after 2 weeks without eating anything.
I then decided to change (finally!) his tank... I bought a 2 gallons tank, with a heater, a filter, etc... My Betta didn't eat/move much more (he never seemed to fully recover from his "disease"). But when I changed his tank, I didn't wait the 3-4 weeks for the nitrate cycle (I just learned from it...). I use water (I wait 48h before use the water, and put some conditionner).
Finally, I have a couple of questions :
- Is it that wrong, for the nitrate cycle? If so, what can I do?
- 2 G. Tank, with heater/filter... what should be the frequency of my water changes?
-What can I do for my Betta? He doesn't seem to have any physical scars...
PS. He always stay on top... doesn't move a lot...
Thanks a lot and sorry for the long message (my 1rst) and the mistakes/grammars (english isn't my first language...).