Too big of steps in limits theorem 1 and limits example 1

[bluelightning32]

I took me a while to understand the proof of limits theorem 1 (in limits.html), because the steps it takes are a little too large.

Instead of:
|w_0 - w_1| <= |w_0 - f(z)| + |f(z) - w_1| < 2 * epsilon

The following would have been easier to follow, given that the proof says that 2 * epsilon = |w_0 - w_1| and |f(z) - w_0| < epsilon and |f(z) - w_1| < epsilon:
2*epsilon = |w_0 - w_1| = |(w_0 - f(z)) + (f(z) - w_1)| <= |w_0 - f(z)| + |f(z) - w_1| < 2 * epsilon

In example 1 right below that, I found the following step hard to follow (given that 0 < |z - z_0| < delta = 1):
|z^2 - z_0^2| = |z - z_0||z + z_0| < |z-z_0 + 2z_0| < |z-z_0| + 2|z_0| < 1 + 2|z_0|

Adding one step in the middle would better point out where the delta inequality was used:
|z^2 - z_0^2| = |z - z_0||z + z_0| < |z + z_0| = |z-z_0 + 2z_0| < |z-z_0| + 2|z_0| < 1 + 2|z_0|

The text then follows with, "now, if delta = (epsilon / (1+ 2|z_0|)), then 0 < |z-z_0| < delta implies that |z^2-z_0^2| < 1 + 2|z_0| < epsilon."

That's confusing and incorrect. The statement is false when epsilon=0.1, z_0=1, and z=0.99. It simplifies as:
|z^2-z_0^2| < 1 + 2|z_0| < epsilon
|(0.99)^2 - 1^2| < 1 + 2*1 < 0.1
|.9801 - 1| < 1 + 2 < 0.1
0.0199 < 3 < 0.1

Instead, I suggest moving the delta = min{1, epsilon / (1+ 2|z_0|)} earlier, and using it directly in the earlier inequalities.
|z^2 - z_0^2| = |z - z_0||z + z_0| < delta * |z + z_0| = delta * |z-z_0 + 2z_0| < delta * (|z-z_0| + 2|z_0|) < delta * (delta + 2|z_0|) <= delta * (1 + 2|z_0|) = (epsilon / (1+ 2|z_0|)) * (1 + 2|z_0|) = epsilon