What was found
REPOSITIONING FLIGHT KAPF-KISM, THE CAPTAIN IS THE PILOT MONITORING, AND THE FIRST OFFICER IS THE PILOT FLYING. DURING THE CLIMB OUT AT AN ALTITUDE OF 10,000 FT, THE CREW RECEIVED AN APU FIRE INDICATION. PM (CAPTAIN) FOLLOWED THE IMMEDIATE CHECKLIST ITEMS AND PUSHED THE APU FIRE BUTTON FOLLOWED BY THE EXT 2 BUTTON TO DISCHARGE THE #2 FIRE EXTINGUISHER INTO THE APU. APU FIRE INDICATION CEASED AND THE APU WAS SHUT DOWN BECAUSE OF THE APU FIRE BUTTON PUSH. THE CREW DECLARED AN EMERGENCY AND DIVERTED TO KRSW WHICH WAS RIGHT BELOW THE AIRPLANE'S FLIGHTPATH. CONTINUED FOR A SAFE LANDING ON RUNWAY 06 AT KRSW. UPON FURTHER INSPECTION OF THE AIRPLANE BY THE FLIGHT CREW AND AIRPORT FIRE CREWS, NO PHYSICAL INDICATIONS OF FIRE WERE PRESENT.