Exercise 1565Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

Countable union of countable sets

#countable union#diagonalization#countable

Statement

Show that a countable union of countable sets is countable.

Hint

Enumerate the sets $A_n=\{a_{n,1},a_{n,2},\ldots\}$ ; enumerate pairs $(n,k)$ diagonally.

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Solution

Enumeration. For each $n$ , enumerate $A_n=(a_{n,k})_{k\ge1}$ . The diagonal enumeration of $\mathbb{N}^2$ gives a surjection $\mathbb{N}\to\bigcup_n A_n$ .

Countable. Image of a countable set under a function is countable.

Conclusion. $\bigcup_n A_n$ is countable.

Countable union of countable sets

Statement

Solution

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