FOLLOW AFTER

The word obsequious, meaning "excessively obedient", was borrowed in the late fifteenth century from Latin obsequiem, which pretty much meant the same thing but in a less pejorative sense. That's from the prefix ob-, here meaning "after" (from Proto-Indo-European epi, "against"), and the verb sequi, which meant "to follow" in Latin. This is the same root as the one we see in non sequitur (something that does not follow), consecutive (something that follows together), suitor (one who follows another), sequel (something that follows), and many more similar words. Finally, etymologists reconstruct sequi to the Proto-Indo-European root sekw, which also just meant "to follow". Since its introduction into the English language, obsequious​ peaked in usage in 1779 and has been on the decline since.