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 \(S_{n}\ \text{is countable so}\ T_{n}\) is non empty.
so, there exist a sequence \(t_{n} \in T_{n}\)
define, \(f:S \rightarrow N \times N\) be the mapping defined by
\(f(x) = (n, t_{n}(x))\)
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.
\(N \times N\) is countable
so, there exist a: \(N \times N \rightarrow N\)
since composite of injections is injection thus,
a circ \(f:S \rightarrow N\) is an injection
therefore, S is countable