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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. The integration by parts formula may be stated as What i often do is to derive it from the product r. Aubin, un théorème de compacité, c.r

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). When can we say a multiplicative group of integers modulo $n$, i.e., $u_n$ is cyclic $$u_n=\\{a \\in\\mathbb z_n \\mid \\gcd(a,n)=1 \\}$$ i searched the internet but.

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