BA3100 CREW AVOIDS NMAC WITH LEAR JET THANKS TO TCASII.

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WHILE CLBING THROUGH 15500 FT; I NOTICED PROX TFC AT OUR 6 O'CLOCK AND 5 NM (ON TCASII). TCASII WAS SET TO 12 NM RANGE AND 'ABOVE' FOR THE CLB. THE TFC WAS 600 FT LOWER AND CLBING. AS WE REACHED 16000 FT; I ASKED ATC ABOUT THE TARGET. AS ATC RESPONDED 'NO OBSERVED TFC;' THE TCASII ISSUED AN RA TO DSND AT 800-1000 FPM. THE COPLT PITCHED DOWN; COMPLYING WITH THE RA AND I NOTIFIED ATC WE WERE FOLLOWING AN RA AND THAT WE HAD TFC IN SIGHT 200 FT ABOVE US. IT WAS A LEAR JET AT APPROX 16000 FT OFF OUR R SIDE. WE HAD DSNDED OUT OF 16000 FT AND SAW HIM. ATC THEN ISSUED A TFC CALL FOR 'FASTER MOVING TFC' AT OUR 1 O'CLOCK. AS WE PASSED THROUGH 15500 FT; THE TCASII ANNOUNCED 'CLR OF CONFLICT.' WE LEVELED AT 15400 FT AND INITIATED A CLB BACK TO 16000 FT WHILE NOTIFYING ATC. THE LEAR JET LEVELED AT 16500 FT; AND CHKED IN WITH ATC. IN MY ESTIMATION WE WOULD HAVE EITHER COLLIDED WITH THIS LEAR JET OR MISSED BY 20 FT HAD WE NOT DSNDED ON THE RA. AT FAULT WAS THE LEAR JET OVERTAKING US AND NOT COMPLYING WITH THE SEE AND AVOID REQUIREMENT. VISIBILITY WAS AT LEAST 60 NM AND THERE WERE NO CLOUDS. ATC'S FAILURE TO MONITOR AND NOTIFY US OF PROX TARGETS CONTRIBUTED TO THE SIT. I DO NOT FEEL MY CREW COULD HAVE POSSIBLY DONE ANYTHING DIFFERENT. THE LEAR JET WAS BEHIND US; OVERTAKING US; NOT TALKING TO ANYONE; WE PICKED HIM UP 5 NM BEHIND US; WE FOLLOWED AND COMPLIED WITH ALL PROCS AND REGS.