Added: Interpreting the verb "to motivate" in another way, I always discuss the history of the ideas in some depth (I learnt it myself from wikipedia and books on the history of mathematics), and just how much people struggled to find the "right" definition, with no success, until Bernard Bolzano (primarily a Catholic priest!) finally hit upon an idea that worked in 1810. What idea were they trying to capture? Why was it so hard? How come it took 2500 years (Zeno of Elea to Bolzano) to find the right idea?
I'll also discuss the definition having been reworked and distilled by many many people- first its inventors, then mathematicians, then textbook writers, becoming more and more refined and smaller and smaller until that which is left looks to one who sees it for the first time like a small cold hard stone. It's only once you polish it (working it over in your mind, and solve problems) and shine it under a bright light (make sense for yourself of all those nested quantifiers) that you can finally see it for what it is- a diamond.