Talk:Quantifier shift

Examples from calculus
In some metric space (M,d), let B(a,r) be the (metric) ball with center a and radius r.


 * Convergence of $$(x_n)_{n\in\N}\subset M$$ is
 * $$\exists a\in M\forall \epsilon>0 : x_n\in B(a,\epsilon)\mbox{ for almost all }n$$


 * vs.


 * Cauchy-criterion (equivalent to the usual definition):
 * $$\forall \epsilon>0\exists a\in M : x_n\in B(a,\epsilon)\mbox{ for almost all }n$$


 * Continuity on a set $$U\subset M$$ vs. uniform continuity on U is the better known example
 * $$\forall x\in U \forall \epsilon>0\exists\delta>0: f(B(x,\delta))\subset B(f(x),\epsilon)$$


 * vs.


 * $$\forall \epsilon>0\exists\delta>0\forall x\in U : f(B(x,\delta))\subset B(f(x),\epsilon)$$

--LutzL (talk) 16:58, 25 December 2012 (UTC)