Why did my betta fish die?

He was very lively when we first bought him home. The people at Petco said that he would not need a heater and he really didn't even need the filter that came with the 2.5 gallon tank. We did everything we were told:treat the water with the conditioner and let it run through the filter for 24 hours before putting in the fish; feed him only about 3 times a week or every other day; do a 50% water change 1 a week; don't keep the tank in direct sunlight. He seemed ok for about 2 weeks then I noticed he wasn't really swimming much. He would stay at the bottom of the tank not moving but was still alive. Every once in a while it looked like he took a deep breath. He would not eat. I went to another Petco and tried to buy a heater and they said the heater would kill the betta, so I did not buy it. They told me the betta didn't do much, so that was normal. Then he started laying on his side. He still looked normal (no fins falling off or stuff like that). I read online on various sites that he SHOULD have a heater and also that if he had labored breathing and erratic swimming that he might have an infection. Most of the bloggers agreed that some salt would clear up infections. So, I bought a heater anyway and some aquarium salt. I followed the directions and mixed the salt separately in another bowl (using the water he was already in) and connected the heater. I also attached a thermometer to the tank and it read about 70 or 72 and after the heater about 74. He actually started to sit up (instead of laying on his side, he was now holding himself up regularly. He even responded to me when I called him like when I first bought him. He swam around a little bit and went hiding in his little castle. I thought all was better now, but then this morning he was just laying at the bottom of the tank, this time not moving at all. Does anyone know what is was that killed him? It makes me think that ADDING the heater killed him, but it seemed like he was a little happier when it got warm. Maybe the salt? Maybe he starved? I was still trying to feed him little bits of food, but he never went up to get them. I do not want to buy my son another betta (he wants one), because this is the second one that died. The first one I think was because of too small a tank. We had bought the "betta" tank they sell that is small with a divider and just took out the divider to let him swim freely. But he was not lively when we bought him and the water was cloudy after a few days. I didn't even get the chance to change that one's water because when I got home after work, it was already dead in only 2 or 3 days. I guess what I really need to know is, does the betta actually need a heater or not? too many people differ on this and it just seems to me that the water was ice cold (it is very cold where I live). We are very sad...