Well, that's what most pilots have been trained to do, and it's what airmanship dictates. Unfortunately some obviously either forget or 'don't think it's important.' This seems to be the case here. Believe me, usually, passing the traffic info should prevent this.
It's odd that the other aircraft didn't have an RA though, if his TCAS system is switched on, the RA should be coordinated and both aircraft will get one.