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 $(x_{n})_{n\in \mathbb {N} }\subset M$ 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)