Applying Cantor's diagonalization technique to sequences of functions

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SUMMARY

The discussion focuses on applying Cantor's diagonalization technique to construct a nested sequence of functions that converges at every point in a specified set A. The participants emphasize the importance of visual aids, such as diagrams, to clarify the construction process. Additionally, there is a query regarding the necessity of using induction on two variables, m and k, to facilitate the proof. The conversation highlights the collaborative nature of problem-solving in undergraduate analysis.

PREREQUISITES
  • Understanding of Cantor's diagonalization technique
  • Familiarity with nested sequences of functions
  • Basic knowledge of mathematical induction
  • Proficiency in LaTeX for mathematical notation
NEXT STEPS
  • Study the application of Cantor's diagonalization in functional analysis
  • Explore the properties of convergence in nested sequences of functions
  • Review techniques for mathematical induction involving multiple variables
  • Practice constructing diagrams to illustrate complex mathematical concepts
USEFUL FOR

Undergraduate mathematics students, educators in analysis, and anyone interested in advanced mathematical proofs and techniques.

jdinatale
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As usually, I type the problem and my attempt at the solution in LaTeX.

6214a.png


6214b.png


Ok, so for the last part (c), I obviously have the diagram down, now I just have to construct the nested sequence of functions that converges at every point in A. I drew a diagram to help illustrate the idea.

picture.jpg


Would I need to do induction on two variables? (m and k)
 
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Bump. This stuff is only undergraduate analysis, I know someone can help.
 
I think I might have it, let me know what you guys think:

dat.png
 

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