First of all get a volt meter on the battery when the car is running, this should show that amount of volts the alternator is giving out should be 14.2volts.
Car Not running
Discharged State:
12.5v
Charged State:
12.7v
Car running (Charging)
Charging
14.2v
Loaded Battery (lights, Rear demister, heater, etc)
12.7volts
Lead acid car batteries are not very good in the cold, the acid tends to get very weak and can freeze giving a poor cell, this causes the cell to die over a short period of time.
Sounds to me that you have a weak battery, it happened to myself last week, managed to catch the car and run, but all the warning lights stayed on!!!!, got a jump from the wifes car and it went no probs.
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