Egorov's Theorem - Generalizations - Korovkin's Version - Proof

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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