00:01
Now to prove that the union of denumberably many sets bk, each of which is countable, we have to show that their union is also countable.
00:18
Now, let bk, k belongs to n be a sequence of countable sets, be a sequence of countable sets and let b is equal to union.
00:34
Union k belongs to n bk.
00:38
For all n belong to n, let fk denote the set of all injections from bk to n.
00:58
Now since bk is countable, since bk is countable, since bk is countable, so f k is non -empty using axiom of countable choice there exists sequence f k belongs to n such that f k belongs to capital f k for all n belongs to n now let phi be a mapping from b to n cross n different defined as phi of x is equal to k f k x where k is the k is smallest natural number such that k is the smallest natural number such that k belongs to b k.
02:08
From well ordering principle, such a k, such a k exist and hence the mapping exists, mapping and hence the mapping, phi exist.
02:30
Since each, since each fk is an injection.
02:37
So it is trivial that phi is also an injection.
02:41
5 is also an injection...
