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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
- Newest Questions - Mathematics Stack Exchange
Mathematics Stack Exchange is a platform for asking and answering questions on mathematics at all levels
- 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
- Double induction example: $ 1 + q + q^2 + q^3 + \cdots + q^ {n-1} + q^n . . .
Slightly relevant: you can see my answer on this thread for a proof that uses double induction (just to get you exposed to how the mechanics of a proof using double induction might work)
- 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
- modular arithmetic - Prove that that $U (n)$ is an abelian group . . .
Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian
- geometry - Circle revolutions rolling around another circle . . .
At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this? Consider Circle A rolling along a straight line the length of the circumference of Circle B Then it will revolve 3 times
- Mnemonic for Integration by Parts formula? - Mathematics Stack Exchange
The Integration by Parts formula may be stated as: $$\\int uv' = uv - \\int u'v $$ I wonder if anyone has a clever mnemonic for the above formula What I often do is to derive it from the Product R
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