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Probably the battery if it's an old one (5 years or more).
Maybe the alternator.
If you have a voltmeter you can test it without going to a mechanic and replacing the battery.
Measure the voltage of the battery with engine off, if it's below 12V (even 11,8V) then your battery is drained or dead.
Then start/jumpstart the car, and measure the voltage again. If the alternator is working fine you should now get a voltage reading of about 14V, which means the battery is charging. If you don't get a voltage increase then you can stop looking, your alternator is dead. If you do, you need to drive about 30 minutes at 40mph, driving 10 minutes is not enough to fully charge a battery. If the car alternator is up and running then a fully charged battery should indicate 13.7-14.7 volt when measured.
If you shut down your car after that 30 minute drive and the battery's charge drops below 12,5(ish)V, your battery is faulty/dead.
So maybe the cold weather and driving short distances drained the battery faster than it could recharge. Maybe both battery and alternator are fine. Only way to know is to test it
Maybe the alternator.
If you have a voltmeter you can test it without going to a mechanic and replacing the battery.
Measure the voltage of the battery with engine off, if it's below 12V (even 11,8V) then your battery is drained or dead.
Then start/jumpstart the car, and measure the voltage again. If the alternator is working fine you should now get a voltage reading of about 14V, which means the battery is charging. If you don't get a voltage increase then you can stop looking, your alternator is dead. If you do, you need to drive about 30 minutes at 40mph, driving 10 minutes is not enough to fully charge a battery. If the car alternator is up and running then a fully charged battery should indicate 13.7-14.7 volt when measured.
If you shut down your car after that 30 minute drive and the battery's charge drops below 12,5(ish)V, your battery is faulty/dead.
So maybe the cold weather and driving short distances drained the battery faster than it could recharge. Maybe both battery and alternator are fine. Only way to know is to test it