Proving the limit of sinx/x as x approaches 0 equals 1

[Terrell]

this is a geometric proof from James Stewart's calculus textbook page 192. I'm confused in the sequence of inequalities as part of the proof...
theta = arcAB < AB + EB ==> arcAB < AE + ED. How did EB turned into ED?
please check the picture I've uploaded with this post

 

[QuantumQuest]

In the first diagram, it' easy to see that $\left|EB\right| < \left|ED\right|$, so what you ask follows. Then there are just substitutions.

 

[RUber]

Point D was described in the first half of the proof. It should be clear that |ED| > |EB|.
The benefit of doing the substitution is that you can reduce the expression to one involving straight segments related to theta by trigonometric functions.
$\tan \theta = \frac{|AD|}{|OA|}$