Q: Need help determining if alternator is bad

My car has had a battery for 5 years. It began cranking very slowly, but never failed to start the car. I decided to replace the battery with a new one. The parts store gave me one with much higher CA,CCA, and RC, compared to the old one. It took only 2 days after installing it when the battery light turned on in the dashboard. I went back to the old battery to see if the battery light turns off. The car turned on and then suddenly the rpm began decreasing. I tried to accelerate it to keep it on but did not work.The radio turned off saying error on car's computer, the headlights turned off. Now the car wont even crank, basically the battery was dead and the alternator has failed. My question is did the battery that the parts store gave me, ended up overworking and damaging my alternator or was the alternator prone to failing sooner or later? Before the new battery my car worked fine. It had slow cranks because of the age of battery but never failed me and never had the battery light on

My car has 182863 miles.
My car has an automatic transmission.