{"articles":{"limit-supremum@set-theory":{"content":"<p>The limit supremum is the set of points that reocurr. Possibly these points are cyclic or oscillating. \\[{x \\in{} {{\\operatorname{LimSup} A}} \\text{ if } \\forall{} n \\exists{} m \\ge{} n, x \\in{} {A}_{m}}\\] <br> \\[{x \\in{} {{\\operatorname{LimSup} A}} \\text{ if } x \\in{} {{{\\bigcap}}_{n}}^{{1:\\infty}} {{{\\bigcup}}_{m}}^{{n:\\infty}} {A}_{m}}\\] <br> \\[{x \\in{} {{\\operatorname{LimSup} A}} \\text{ if } x \\in{} {{\\underset{{n \\to{} \\infty}}{\\operatorname{Lim}} {{{{\\bigcup}}_{m}}^{{n:\\infty}} {A}_{m}}}}}\\] <br> We can think of the limit supremum as starting with the supremum, then as \\(n\\) grows, the points that will never reocurr again are gradually removed.</p>","names":[[["Limit supremum",""]," ","of"," ",["a sequence of sets","set-sequence"]]]}},"parameters":["@"],"style":"Notion"}