2001 Ford F-150 battery/alternator/starting

I just recently purchased 2001 f150 bought it drove it, parked it overnight, next morning started but lights dimmed, wiper slowed radio shut off, 1st alternator didnt work, 2nd work drove it 5 hrs straight parked it sat overnight morning dead again, boosted it mechanic said battery needs to be charged or a new one put in, put a new one in, drove all day fine started several times no problem, on the way home radio shuts off lights dim and battery guage reads lowest level, get home idles fine, shut truck off try starting nothing, what could this be? Is it that I should have replaced the alternator after replacing the battery or wiring from alternator out to starter> I have had it two days I am real fustrated.