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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.

Thanks!

[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.

Thanks!

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