Welcome, sorry to hear about your little guy.. lets see if we can help some :)
What temperature is your tank? 72 degrees F That is pretty low for a betta. They are tropical fish and it would do wonders with a heater. I suggest nothing lower then 76*F, up to 82*. What happens when a betta is in too cold of water for a long time, they become lethargic, their metabolism slows and so does their eating- which in turn can cause health issues. It's best to find a small heater 10-15 watt should be fine to keep him at the appropriate temp.
What type of food do you feed your betta fish? flakes
How often do you feed your betta fish? once a day
Are they betta specific flakes? Tropical flakes normally doesn't meet the nutritional requirements for bettas. Fish/meat meal should be the first ingredient, which a high crude protein %. Flakes are notorious for fouling up the water if uneaten ones aren't removed right after feeding. As well as sometimes they can cause the betta to bloat up. You can feed them betta specific flakes, but I would mix it up with pellets, as they are a very good staple. Also live and frozen foods are wonderful.
Bettas do best with small, multiple meals rather then just one. With one meal you run the risk of either under feeding, or over feeding which causes trouble either way. Even if you just do breakfast and dinner, that would be more idea. Just make sure to not over feed during those meals.
How often do you perform a water change? weekly
What percentage of the water do you change when you perform a water change? 25% to 50%
I personally would up it to 2 50% weekly since it's a smaller tank, one 50% water only, the other 50% siphoning out the substrate (or stir it up and dip out the waste with a cup).
What you most likely saw was the fungus version of fin rot. Fin rot will happen when a fish tears his fin (doesn't matter how he does it, but just as long as it's damaged) and the water isn't all that clean. By upping the water changes a week as I suggested it should take care of the problem itself. If you notice the edges getting black and flaking off, or more white cottony like substance grows on his fins, then you could start off on AQ salt treatments.
1 teaspoon of AQ salt per gallon (dissolve it prior to adding him in it), with your regular water conditioner, daily 100% water changes for no longer then 10 days. That should kill off the bacteria that causes fin rot.
If I miss anything, I'm sure OFL will be able to fill you in on more :) She's the expert! Good luck to you.