Why not number some definitions in Rudin's book?

Question

I'm reading Rudin's Principle of Math Analysis

Why some definitions about functions in Chapter 8, there are no the number marks, and the boldfaced words Definition?

This chapter only have 22 marked theorem and definitions and...

For example

Section 3

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We define

$$\begin{align*}E(z)=\sum _{n=0}^{\infty } \frac{z^n}{n!}\quad \quad\tag{1}\end{align*}$$

Section 4

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Let us define

$$\begin{align*}C(x)=\frac{1}{2}[E(\text{ix})+E(-\text{ix})]\tag{2}\end{align*}$$

Is there some reason's in doing these things? i.e. Boldface and number some theorems/definitions in one book.

Answer 1

These are purely local definitions, used only temporarily. The whole point of those sections is to prove that this function $E(z)$ is just the exponential function $e^z$, that this $C(x)$ is just $\cos x$, and that the similarly defined $S(x)$ is just $\sin x$.