# Proof

• Since each $$D_n$$ is countable by hypothesis, we can write $$D_n=\{x_{nm},~m\in\mathbb N\}$$.
• We claim, that the union $M:=\bigcup_{n\in\mathbb N} D_n$ is again countable.
• In order to see it, we write the elements of the union in a table: $\begin{array}{ccccccc} \left(x_{00}\right)_0,&\left(x_{01}\right)_1,&\left(x_{02}\right)_3,&\left(x_{03}\right)_6,&\left(x_{04}\right)_{10},&\cdots\\ \left(x_{10}\right)_2,&\left(x_{11}\right)_4,&\left(x_{12}\right)_7,&\left(x_{13}\right)_{11},&\cdots\\ \left(x_{20}\right)_5,&\left(x_{21}\right)_8,&\left(x_{22}\right)_{12},&\cdots\\ \left(x_{30}\right)_{9},&\left(x_{31}\right)_{13},&\cdots\\ \left(x_{40}\right)_{14},&\cdots\\ \vdots \end{array}$
• The outer index defines an injective function $$f:M\mapsto\mathbb N$$. Therefore, $$M$$ is countable.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983