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 ...
Prove that the sequence $\ {1, 11, 111, 1111, .\ldots\}$ will contain two numbers whose difference is a multiple of $2017$. I have been computing some of the immediate multiples of $2017$ to see how
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...
17 Un U n is cyclic iff n n is 2 2, 4 4, pk p k, or 2pk 2 p k, where p p is an odd prime. The proof follows from the Chinese Remainder Theorem for rings and the fact that Cm ×Cn C m × C n is cyclic iff (m,n)= 1 (m, n) = 1 (here Cn C n is the cyclic group of order n n).
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 ...
I'm trying to proof the following isomorphism $$U (n) \simeq \frac {SU (n) \times U (1)} {\mathbb {Z}_ {n}}$$ So I'm using the first Isomorphism theorem: http://en ...
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.
