Please let me know if the following references/sites are helpful to you:

Solving Mathematical Problems by Terrance Tao

www.albertstam.com/Solving_Mathemacal_Problems_by_Terrance_Tao.pdfCached

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Solving Mathematical. Problems. A Personal Perspective. Terence Tao. Department of .... PRooF. Call the triangle ABC. Now let P be the intersection of the.

1.  Mathematical Thinking: Problem-solving and Proofs - IUUK

iuuk.mff.cuni.cz/~andrew/MTPSP.pdfCached

Mathematical. Thinking. Problem-Solving and Proofs. Second Edition. John P. D' Angelo. Douglas B. West. University of Illinois — Urbana. PRENTICE HALL.

2.  Solving Mathematical Problems by Investigation - CiteSeerX

citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.577.1616&rep=rep1...pdfCached

by JBW YEO - ‎Cited by 13 - ‎Related articles

solving closed mathematical problems by investigation is a contradictory notion. ... investigation can help them to develop a more rigorous proof for their.

3.  TEACHING PROBLEM-SOLVING SKILLS - Mathematical Association ...

https://www.maa.org/sites/default/files/pdf/upload_library/.../Schoenfeld794-805.pdfCached

by AH SCHOENFELD - ‎Cited by 170 - ‎Related articles

mathematicians develop consistent and useful problem-solving strategies. Second: most ... For example, consider the following problems. Problem I: Let a, b , c, ...

Dennis

Dennis Mazur

 Dear Dennis

Thank you for your reply.

All of your materials are very interesting.

It has a wide coverage and is too extensive for proof of my specific problem.

I will read them when I have time in the future.

After posting the question, I advanced a little.

Instead of proving directly that the limit value of any convergence sequence in the subsets E is contained in E, I have switched to prove that closure of some sub set of E coincides with E.

In this proof, I use the Lemma cited by the author, so I think I am getting closer to the correct answer.

Best regards

Mashiro Fujimoto

The Lemma  proposed by the author guarantees that A is a subset of E. It is more a suggestion to use, to prove iii).

Dear Jorge

Thank you for your reply.

Prove (iii) does not need the Lemma.

For any $f \in L_0^ + ,\;s.t.\;f \le h,\;h \in E$, $f$ satisfies the condition of the definition of $E$.

$A \subset E$ is obvious because if $h \in E$, $h$ satisfies the condition of the definition of $E$ as ${h_n} = h$.

Best regards.

Mashiro Fujimoto

It is clear, that your question concerns a theorem in a paper by Walter Schachermayer, a  professor in probability and quantitative finance in Vienna. My suggestion would be to ask professor Schachermayer for advise. By the way, in always pays to visit Schachermayer's homepage . Kind regards, J.W.N.

The Lemma is another formulation of that of Komlos, which can be used to prove the Doob-Meyer decomposition of a submartingale. This is a result that generalize the Bolzano-Wierstrass result for bounded sequences in R^n.

To prove that E is closed you just need to proof that the limit of every sequence is also in E, because of this Lemma you can always extract a subsequence that converges to something in A and so the limit of the sequence in E.

Dear  Alejandro Santoyo

Thanks for your reply.

The attached file is my reply.

Best regard.

Masahiro Fujimoto