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Math Help - Family of monotonic strictly increasing functions...

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Family of monotonic strictly increasing functions...

    Definitions:

    1. For functions f,g:NN, we say fg if there exist nN such that for nm we have f(m)g(m)
    2. Family F of functions from N to N is unbounded if for every function g:NN, exist fFsuch that fg isn't holds.

      Question:

      F
      is unbounded family of monotonic
      strictly increasing functions from N to N. Show that for everyg:N→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!

      NNN
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  2. #2
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    Re: Family of monotonic strictly increasing functions...

    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)
    Last edited by Link; November 24th 2012 at 02:47 PM.
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