Family of monotonic strictly increasing functions...

Dec 2009
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434
Russia
Definitions:

  1. For functions [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]:[/FONT][FONT=MathJax_AMS]N[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_AMS]N[/FONT], we say [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math]g[/FONT] if there exist [FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_AMS]N[/FONT] such that for [FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math]m[/FONT] we have [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Main])[/FONT]
  2. [FONT=Georgia, Times New Roman, Times, serif]Family [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_Math]F[/FONT][FONT=Georgia, Times New Roman, Times, serif] of functions from [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_AMS]N[/FONT][FONT=Georgia, Times New Roman, Times, serif] to [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_AMS]N[/FONT][FONT=Georgia, Times New Roman, Times, serif] is [/FONT]unbounded[FONT=Georgia, Times New Roman, Times, serif] if for every function [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_Math]g[FONT=MathJax_Main]:[/FONT][FONT=MathJax_AMS]N[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_AMS]N[/FONT][/FONT][FONT=Georgia, Times New Roman, Times, serif], exist [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_Math]f[FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math]F[/FONT][/FONT][FONT=Georgia, Times New Roman, Times, serif]such that [/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_Math]f[FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math]g[/FONT][/FONT][FONT=Georgia, Times New Roman, Times, serif] isn't holds.[/FONT]

    Question:

    F
    is unbounded family of monotonic
    [FONT=Georgia, Times New Roman, Times, serif]strictly[/FONT] [FONT=Georgia, Times New Roman, Times, serif]increasing functions from N to N. Show that for everyg:N[/FONT][FONT=MathJax_Main]→N and infinite set X(subset of N) exists f in F such that g(n)<f(n) for infinite n in X.


    I even don't know from where to start...


    Thank very much! [/FONT]

    [FONT=MathJax_AMS]N[FONT=MathJax_AMS]N[/FONT][/FONT][FONT=Georgia, Times New Roman, Times, serif][/FONT][FONT=MathJax_AMS]N[/FONT]
 
Nov 2012
39
10
Hyrule
Hope this is correct...

First, notice that saying f≤∗g isn't holds means there exists infinite values y0,y1... such that g(yn)<f(yn).
For each n, let x(n) be the smallest x greater than n. the image of n -> x(n) is an infinite set.
we define h like this : h(n)=g(x(n))
We know that there exist f in F such that f(y0)>h(y0); f(y1)>h(y1).... (yi are values in N)

since f(y0)>h(y0) then f(y0)>g(x(y0)) AND we have x(y0)≥y0, plus we know that f is increasing, so f(x(y0))≥f(y0)>g(x(y0)), so we proved that f(x(y0))>g(x(y0)), and we can do the same thing for all x(yn) wich are infinitely many

(sry for mistakes english isn't my native language)
 
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