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Negation of this statement

  1. Jan 30, 2008 #1
    I am stuck in trying to take the negation of this statement:

    [tex](\forall \varepsilon>0)(\exists N \in N)(\forall n,m\geq N)(\forall x \in R [|f_n(x)-f_m(x)|< \varepsilon][/tex]

    One of my thoughts was that in order to move the negation inside the brackets, all I need to do is say [tex](\exists \varepsilon \leq 0)[/tex]...and everything else remains unchanged.

    However, my other thought was to somehow move the statement [tex]\varepsilon > 0[/tex] to the end of the original statement and make it: [tex](\varepsilon > 0 \Rightarrow |f_n(x)-f_m(x)|< \varepsilon)[/tex]

    If you can help me in anyway, it would be greatly appreciated.

    Last edited: Jan 30, 2008
  2. jcsd
  3. Feb 1, 2008 #2


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    It seems like you need something like there exists an eps >0 for which there is no N for the property (a Cauchy sequence in the sup norm topology).
  4. Feb 1, 2008 #3
    I figured it out...here is the solution for anyone who is curious.

    [tex](\exists \varepsilon>0)(\forall N \in N)(\exists n,m\geq N)(\exists x \in R [|f_n(x)-f_m(x)| \geq \varepsilon][/tex]

    Thanks for the help!
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