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(Un-)Countable union of open sets - Mathematics Stack Exchange A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in this case, the union or intersection of any finite collection of open sets is open) the validity of the property for an infinite collection doesn't follow from that In other words, induction helps you prove a
If a series converges, then the sequence of terms converges to $0$. @NeilsonsMilk, ah, it did not even occur to me that this involves a step See, where I learned mathematics, it is not unusual to first define when a sequence converges to zero (and we have a word for those sequences, Nullfolge), and only then when a sequence converges to an arbitrary number, by considering the difference
study of the sequence (Un) defined by $U_ {0}=a$ and $U_ {n+1}=a+\frac . . . Show that (Un) is bounded, convergent and find its limit To prove that the sequence is bounded i intuitively used the fixed point theorem because at first glance i don't really know the appropriate way to study this sequence as it's neither arithmetic nor geometric or the both
functional analysis - Where can I find the paper Un théorème de . . . J P Aubin, Un théorème de compacité, C R Acad Sc Paris, 256 (1963), pp 5042–5044 It seems this paper is the origin of the "famous" Aubin–Lions lemma This lemma is proved, for example, here and here, but I'd like to read the original work of Aubin However, all I got is only a brief review (from MathSciNet)