AN INBOUND DSNDING B737-400 PULLS UP TO AVOID A C172 THAT WAS SUPPOSED TO MAINTAIN VISUAL SEPARATION FROM THE B737.

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DSNDING ON VECTORS FROM 4000 FT TO 2000 FT; WE WERE POINTED OUT AS TFC TO THE VFR CESSNA. HE CALLED US IN SIGHT AND SAID HE WOULD FOLLOW US. HE DIDN'T! WE PICKED HIM UP VISUALLY ABOUT THE TIME WE GOT A TA WHICH QUICKLY CHANGED TO AN RA FOR A 2100 FPM CLB. WE FOLLOWED RA; PASSED TFC; PROCEEDED TO NORMAL LNDG. APCH WAS OFF THE HOOK BECAUSE THE VFR TFC CALLED US IN SIGHT. THE ONLY PROB I HAD WITH IT ALL WAS THAT APCH OFFERED US NO INFO ABOUT WHERE THE TFC WAS IN RELATION TO US OR SEEMED TO HAVE NOTICED HOW CLOSE WE CAME. I KNOW IT WAS AN HONEST MISTAKE AND THE CTLR HAD NO OBLIGATION TO US ONCE THE CESSNA HAD US IN SIGHT; BUT A LITTLE MORE INFO TO US MAY HAVE HELPED US RESOLVE THE SIT MORE QUICKLY AND LESS CLOSELY. SUPPLEMENTAL INFO FROM ACN 426886: I WAS ON APCH CTL FREQ REQUESTING PERMISSION TO BE INBOUND FOR LNDG. I WAS UNDER THE IMPRESSION THAT THE HDG I WAS GIVEN WOULD PUT ME BEHIND THE ACFT THAT I WAS TO FOLLOW TO THE ARPT. I SAW THE ACFT AHEAD AND TO MY R WHICH APPEARED TO BE BELOW MY ALT AND DSNDING -- I MAINTAINED MY ALT. MY ATTN WAS DIVERTED BY A XMISSION FROM ANOTHER RADIO REQUESTING MY ASSISTANCE. I HEARD A XMISSION FROM THE ACFT WHICH I WAS TO FOLLOW THAT HE WAS INITIATING A CLB. I SAW THE AIRPLANE CLBING THROUGH MY ALT AND PASS OVERHEAD BEFORE I WAS ABLE TO REACT. I SHOULDN'T HAVE ASSUMED THAT THE HDG THAT I WAS GIVEN WOULD KEEP ME CLR OF TFC. I SHOULD HAVE BEEN MORE VIGILANT IN AVOIDING THE CONFLICT AND I SHOULDN'T HAVE BEEN DISTR BY THE XMISSION FROM THE OTHER RADIO. FURTHERMORE; THE OTHER ACFT SEEMED FARTHER AWAY THAN WHAT IT ACTUALLY WAS. THEREFORE; IT WAS NOT PERCEIVED AS A THREAT UNTIL IT WAS TOO LATE FOR ME TO REACT.