I'm a high school student interested in higher mathematics. The only resource I've found for analysis, besides the local tech institute, is Youtube and its wonderful collection of lectures. This is the series I'm watching on analysis:

YouTube - Real Analysis, Lecture 10: The Relationship Between Open and Closed Sets
That's the tenth lecture. I don't understand the point of the excessive rigor from 9:49 to 22:04. It seems that the proof can be made directly from the definition of closure. "If a set contains all its limit points, it is called 'closed'. If you take a set S and union it with all its limit points S', then it's closed, as it contains all its limit points. QED." What was the point of a 13 minute and two-chalkboard proof for what could be seen from the definition?

Thanks!