Function growing faster than a sequence of functions

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Discussion Overview

The discussion revolves around the possibility of finding a function f(n) that grows faster than a given sequence of functions f_k(n), where k represents the index of the sequence. The focus is on whether such a function can exist under the condition that the ratio f_k(n)/f(n) approaches zero as n approaches infinity for any k. The conversation includes theoretical considerations and implications of continuity and polynomial density.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the nature of the functions f_k(n) in the sequence.
  • Another participant clarifies that f_k are arbitrary functions mapping natural numbers to natural numbers.
  • A participant initially claims that it is impossible to find such a function f(n) and argues based on the properties of continuous functions and the density of polynomials in continuous functions.
  • This participant suggests that since polynomials are countable, one can construct a sequence of polynomials and argues against the existence of a maximum continuous function by proposing the idea of g + 1.
  • Later, the same participant revises their position, stating that if the functions are "pointwise bounded," then it may be possible to find such a function f(n).

Areas of Agreement / Disagreement

Participants express differing views on the possibility of finding a function that grows faster than the sequence f_k(n). While one participant asserts the impossibility under certain conditions, they later acknowledge a potential pathway if additional assumptions are made. Thus, the discussion remains unresolved regarding the general case.

Contextual Notes

The discussion involves assumptions about continuity and the properties of polynomial functions, which may affect the conclusions drawn. The implications of "pointwise bounded" as an assumption are also noted but not fully explored.

simeonsen_bg
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Hi, I have a sequence f_k(n) of functions N-->N, k=1,2... and I need to find a function f(n) growing faster than any f_k, i.e.

f_k(n)/f(n) --> 0 as n-->infinity for any k.

Is it possible and how can it be done?
 
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Do you know anything about the f_k's?
 
f_k are arbitrary functions from the naturals into itself.
 
Ok, i realized that the following is wrong;

I'm pretty sure that it's not possible. I would first claim that for any function f: N -> N has a continuous extension. Then I would use the fact that polynomails are dense in the set of continuous functions on R. Since the set of polynomals are countable, I would construct a sequence of all polynomails. Then I would go to claim that there is no maximum continuous function (by saying suppose g is a maximum continuous function, then consider g + 1, kind of thing). So if I was given an continuous function, I would consider g + 1, and then find a polynomial that represents it arbitrarily close (at least with in 1/2 distance) and claim that g is indeed not a function that bounds my sequence of functions.

Ok. So now we've shown that in general it's impossible. However, if you add "poitwise bounded" as an assumption, then it is possible.
 
Last edited:

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