M20P AND GLF2 HAD NMAC APPROX 1 MI FROM THE END OF RWY 24 AT MVY.

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ENRTE TO MVY; I WAS TRAINING AN INST STUDENT. I REQUESTED A PRACTICE ILS 24 WITH CAPE APCH. THEY APPROVED THIS AND VECTORED ME ONTO THE LOC ABOUT 5 MI OUT. AT APPROX 2 1/2 MI OUT; THEY HANDED ME OFF TO MVY TWR WITH THE INSTRUCTION OF CIRCLE TO LAND RWY 33. I CONTACTED MVY TWR AND THEY CLRED ME TO LAND ON RWY 33 WITH A L BREAK FOR A R DOWNWIND CIRCLE TO LAND. ABOUT 1 MI FROM RWY 24; I OBSERVED A GULFSTREAM SLIGHTLY TO MY L AT APPROX 400 FT ABOVE ME. I BROKE R AND DSNDED AWAY FROM THE GULFSTREAM AND TOLD THE TWR ABOUT THE CONFLICT AND THEY CLRED ME TO LAND ON RWY 33. I MADE A R 270 DEG TURN AND MADE R TFC TO LAND ON RWY 33. I DON'T THINK ANYONE REALIZED WHAT HAPPENED AND I DIDN'T SAY ANYTHING. APPARENTLY; THAT WAS THE END OF IT. I'M NOT SURE HOW THIS OCCURRED OR HOW I COULD HAVE PREVENTED THIS SIT. I'M GLAD I WAS LOOKING OUTSIDE AS THE STUDENT WAS DOING THE PRACTICE APCH. I SHOULD HAVE RPTED IT TO THE TWR; BUT THE INCIDENT WAS OVER AND I DON'T THINK THE GULFSTREAM PLTS KNEW WHAT HAPPENED OR MVY TWR.