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**Plato** You say that you are reading Rudin. Well notice the statement posted in the first quote is found as theorem 2.10 in Rudin. His proof depends upon $\displaystyle \mathbb{J}$, the symbol he uses for $\displaystyle \mathbb{Z}^+$, being well ordered (every non-empty subset having a first term).

Although Rudin does not use this, many other authors do.

Is $\displaystyle S = \bigcup\limits_n {E_n } $ is the countable union of countable sets then $\displaystyle S = \bigcup\limits_n {F_n } $ where $\displaystyle n \ne m\, \Rightarrow \,F_n \cap F_m = \emptyset $.

In other words, $\displaystyle S$ is the countable union of pairwise disjoint sets, each of which is at most countable.

So $\displaystyle x\in S\, \Rightarrow \, x=f¬_{n,j}\in F_n $ define $\displaystyle \phi :S \mapsto J$ as $\displaystyle \phi(x)=2^n\cdot 3^j$.

It is easy to show that $\displaystyle \phi(x)$ is an injection. So use 2.10 to conclude that $\displaystyle S$ is at most countable.