Negating a Universal Statement in Real Analysis

[bguidinger]

I am stuck in trying to take the negation of this statement:

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

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

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

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

Thanks!

 

[mathman]

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).

 

[bguidinger]

I figured it out...here is the solution for anyone who is curious.

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

Thanks for the help!