Simple, cheap thermostat control

I'm trying to come up with a simple, cheap thermostat control for cycling on a computer fan to keep my wine cooler at around 20*C. I could buy a house thermostat, but they are $15-$25 and big and bulky. I could use an arduino, a transistor, and some kind of temperature sensor, and write thermostat code, but that would be $20 in parts, just for a simple application.

I know you can buy snap switch thermostats like they use in thermoelectric coolers, but I have never seen one set to 20C; they are always for high temperatures...and I haven't found one under $10 anyway.

There must be a cheap, simple way to implement thermostat control. Maybe a diode temperature sensor and a comparator? I happen to have a PT100 I'm willing to use.

I suppose and alarm would be cool, but I can't see dedicating an LCD to this. What I have is a 2-chamber foam cooler with my fermenting beverage in one chamber and a bunch of ice in the other one. A computer fan kicks on to circulate air through the ice to keep the temperature in the fermentation chamber roughly 20C +/- 2C. The way my other one works (using a house thermostat) is it just has an LED wired up to come on with the fan. If you notice the LED is staying on or never coming on, you know there's a problem. That's good enough for this application.