<< Of course your alternator is shot, that is why your battery was dead in the first place. >>
This is not always the case....the battery can be dead for many reasons and usually is not the alternator, keeping a door ajar, headlights left on...even extremely cold temperatures can zap a battery, or a dead cell on the battery itself. Alternator is last resort.
Alternator is not that expensive....but the labor involved in getting to, removing, and replacing will cost you.