Proof
Consider the indexed family of sets whose index set is the set of natural numbers m, defined as follows:
Obiviously
and
therefore there is a natural number m0 such that putting A0,m0=A0 the following relation holds true:
Using A0 it is possible to define the following indexed family
satifying the following two relationships, analogous to the previously found ones, i.e.
and
This fact enable us to define the set A1,m1=A1, where m1 is a surely existing natural number such that
By iterating the shown construction, another indexed family of set {An} is defined such that it has the following properties:
- for all m
- for each m there is a natural km such that for all nkm then for all xAm
and finally putting
the thesis is easily proved.
Read more about this topic: Egorov's Theorem, Generalizations, Korovkin's Version
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