In reply to wercat:
Yes, 'last infinitesimal'. Once the process is over we would say (in a very simple, Anglo-Saxon way) that he/she's 'arrived' and no longer 'arriving'. Only a sloppy writer would prefer to use the word 'stopped', though it could work fine in many contexts. I.e. 'Stop' is much more context-bound. Of course, a good writer will always want to be far less ambiguous than this and would say e.g. that the person has 'pulled to halt (outside the inn, or whatever)', 'entered the pub', 'arrived at the bar', is 'close to the bar', is 'close to the pub', is 'close to the pub door', is 'arriving at the pub', is 'still a few hundred yards away but quite close', etc, etc, etc, etc, etc, etc, etc. I.e we would want to say as accurately as we can exactly what we mean. In ordinary language, that everyone can understand.