Grande Punto 1.4- alternator or battery?

[ConorE]

Hi, bought a 2009 Grande Punto yesterday and an hour after I bought it, it refused to start. After a jump start from a mate I took it for a 15-20 minute drive with no issues then it started again about an hour later. However, this morning the same thing is happening as before where the car simply won't start.

I believe it is the battery as the car didn't stop at all when driving it so the alternator is (I hope) keeping it charged, but any other ideas would be appreciated, thanks.