I'm trying to understand the proof for eccentricity; I can get about halfway through it. The problem I'm having is summarized below

If e<1, you get the equation of an ellipse of the form

where

Eq. 4:

The foci of an ellipse are at a distance c from the center, where

Eq 5:

This shows that

It follows from equations 4 and 5 that the eccentricity is given by

If someone could help explain how is derived, or really what it means I'd be so grateful. I'm using Calculus Early Transcendentals 6th Ed. by Stewart (Ch. 10, Sect. 6).