The word "obsequious" traces its origin back to the Latin word "obsequium," meaning "compliance" or "dutiful service."  This, in turn, comes from "obsequi," which means "to accommodate oneself to the will of another." The Latin root "sequi" (meaning "to follow") is also part of the word, highlighting the idea of following someone's wishes closely. 

According to the Oxford English Dictionary (OED), the word "obsequious" was first used in English in the Middle English period, specifically in the year 1447.  The OED's earliest evidence of its use is found in the writings of Osbern Bokenham, a poet and Augustinian friar

Examples of "obsequious" used in a sentence: 
"He cuddled up to his colleagues, begging for approval - he was obsequious, smart, slippery." 
"Diamond has a gentle, self-effacing style about him without seeming obsequious."
"The waiters came hurrying forward with obsequious smiles on their faces." 
"The service in the restaurant was attentive without being obsequious." 
"Barrow was positively obsequious to me until he learned that I too was the son of a laboring man." 
"He succeeded, however, in reversing his father's obsequious policy towards Russia." 
"If you describe someone as obsequious, you are criticizing them because they are too eager to help or agree with someone more important than them."
"Another characteristic of the de la Gardie government was its gross corruption, which made Sweden the obsequious hireling of that foreign power which had the longest purse." 
"He was the creature of every passing mood or whim, incapable of cool and steady judgment or of the slightest self-control - an incalculable weathercock, blindly obsequious to every blast of passion." 
 

Just the facts:

OBSEQUIOUS Definition & Meaning - Merriam-Webster

OBSEQUIOUS Synonyms: 103 Similar and Opposite Words | Merriam-Webster Thesaurus