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

• Convergence of ${\displaystyle (x_{n})_{n\in \mathbb {N} }\subset M}$ is

${\displaystyle \exists a\in M\forall \epsilon >0:x_{n}\in B(a,\epsilon ){\mbox{ for almost all }}n}$

vs.

Cauchy-criterion (equivalent to the usual definition):

${\displaystyle \forall \epsilon >0\exists a\in M:x_{n}\in B(a,\epsilon ){\mbox{ for almost all }}n}$

• Continuity on a set ${\displaystyle U\subset M}$ vs. uniform continuity on U is the better known example

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

vs.

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

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