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Convergence Questions

  1. Dec 1, 2013 #1
    I'm trying to organize my thought processes about real analysis, using general questions to motivate the theory, in the hopes of using this format for when I study functional analysis or something, so it doesn't feel like 50 new ideas & instead is the modification of previously existing ideas in a new domain - thus I'd like to ask some stupid questions & would love any feedback.

    a) Could one look at the theory of convergence of sequences of numbers as a theory of 'countable convergence'?
    b) Could one look at the theory of convergence of sequences of functions as a theory of 'uncountable convergence'?
    (Even though you are dealing with a sequence from N into a set of functions, the domain of the function is uncountable & the essential quality of these things is the uncountability it seems...)
    c) Can we phrase the theory of limits of functions (over ℝ) in terms of (pointwise?) convergence of sequences of functions?
    (We can phrase the theory of limits of functions in terms of limits of sequences of numbers, i.e. sequential continuity, but since there is a discrete quality to this I think it would be nicer to begin with some form of continuity (in the uncountable continuum sense) & think of it's interplay with limits of sequences of numbers more as a theorem relating uncountablility to countability via continuity - thus can we phrase the ε-δ definition of a limit of a function in terms of convergence of a sequence of functions?)
    d) Could one talk about subsequences as the application of set theory (subsets) to the idea of sequences?
    e) Could one talk about monotonicity of sequences as the application of relation theory (order relations like <) to the idea of sequences?
    f) Could one talk about bounded sequences as the application of topology (norm || via a metric) to the idea of sequences?
    (The point here is to try to motivate the necessity for asking these questions, & to establish a general format for the types of questions you want to ask about topic X, i.e. "now we'll examine the topological aspects of differentiability just as we did when examining sequences", thus something like "the limit laws" becomes the application of algebra to countable convergence etc...).

    See how these go, thanks for reading.
     
  2. jcsd
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