How to negate $\forall\epsilon > 0,\; \exists n_0=n_0(\epsilon)\in \Bbb{N},\ni |a_n-a^{*}| <\epsilon,\;\;n\geq n_0$ This became somewhat a concern to me today, when asked to _negate_ the following definition of convergence below; > $$\lim_{n\to \infty}a_n=a^{*},\;\text{i.e.},$$ > > $$\forall\; \epsilon > 0,\; \exists\;n_0=n_0(\epsilon)\in \Bbb{N},\;\ni |a_n-a^{*}| <\epsilon,\;\;n\geq n_0$$ Please, can anyone show me how to negate the above definition?

Negation "percolates" from outside to inside by changing all quantifiers (even the bounded quantifiers used here): $$ \begin{matrix}\
eg&\forall \epsilon>0,&&\exists n_0\in\Bbb N,&& \forall n\ge n_0,&\
i&& |a_n-a^*|<\epsilon\\\ & \exists \epsilon>0,&\
eg&\exists n_0\in\Bbb N,&& \forall n\ge n_0,&\
i&& |a_n-a^*|<\epsilon\\\ &\exists \epsilon>0,&&\forall n_0\in\Bbb N,& \
eg&\forall n\ge n_0,&\
i&& |a_n-a^*|<\epsilon\\\ &\exists \epsilon>0,&&\forall n_0\in\Bbb N, &&\exists n\ge n_0,&\
i& \
eg&|a_n-a^*|<\epsilon\\\ &\exists \epsilon>0,&&\forall n_0\in\Bbb N, &&\exists n\ge n_0,&\
i&& |a_n-a^*|\ge \epsilon\end{matrix}$$