|Jan30-08, 07:31 PM||#1|
negation of this statement
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.
|Feb1-08, 07:48 PM||#2|
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).
|Feb1-08, 10:06 PM||#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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