# Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 Prove that countable union of countable sets is countable.

Discrete mathematics cardinality using Richard Hammacks
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According to the given information, it is required to prove that countable union of countable sets is countable. Let be the sequence of countable sets. Now define: $S=\bigcup {S}_{n}n\in N$ for all n in N denote T, denote the set of all injections from ${S}_{n}\text{to}N$. since is non empty. so, there exist a sequence ${t}_{n}\in {T}_{n}$ define, $f:S\to N×N$ be the mapping defined by $f\left(x\right)=\left(n,{t}_{n}\left(x\right)\right)$ where n is the smallest unique natural number such that x in ${S}_{n}$ from well ordering principle such an n exist hence mapping f exist. each t_n is an injection, thus it follows that f is an injection. $N×N$ is countable so, there exist a: $N×N\to N$ since composite of injections is injection thus, a circ $f:S\to N$ is an injection therefore, S is countable