{"articles":{"limit-infimum@set-theory":{"content":"<p>The limit infimum is the set of points that reocurr in every step. \\[{x \\in{} {{\\operatorname{LimInf} A}} \\text{ if } \\exists{} n \\forall{} m \\ge{} n, x \\in{} {A}_{m}}\\] <br> Since the inner intersection is an increasing sequence, the outer union is simply there to let us take the limit. \\[{x \\in{} {{\\operatorname{LimInf} A}} \\text{ if } x \\in{} {{{\\bigcup}}_{n}}^{{1:\\infty}} {{{\\bigcap}}_{m}}^{{n:\\infty}} {A}_{m}}\\] <br> Note that the inner intersection is monotone, therefore by a result in this chapter, the limit exists. Therefore the limit infimum may be written as follows, and it may be easier to think about it like this: \\[{x \\in{} {{\\operatorname{LimSup} A}} \\text{ if } x \\in{} {{\\underset{{n \\to{} \\infty}}{\\operatorname{Lim}} {{{{\\bigcup}}_{m}}^{{n:\\infty}} {A}_{m}}}}}\\] <br> Also, we can think of the limit infimum as starting with the infimum, then as \\(n\\) grows, gradually adding on the points that eventually always reocurr.</p>","names":[[["Limit infimum",""]," ","of"," ",["a sequence of sets","set-sequence"]]]}},"parameters":["@"],"style":"Notion"}