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1500 questions
110
votes
5 answers
Pointwise vs. Uniform Convergence
This is a pretty basic question. I just don't understand the definition of uniform convergence.
Here are my given definitions for pointwise and uniform convergence:
Pointwise convergence: Let $X$ be a set, and let $F$ be the real or complex numbers.…
Jeff
- 1,641
- 3
- 17
- 23
110
votes
0 answers
Ring structure on the Galois group of a finite field
Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure of a topological ring isomorphic to…
Martin Brandenburg
- 163,620
110
votes
15 answers
How to prove that exponential grows faster than polynomial?
In other words, how to prove:
For all real constants $a$ and $b$ such that $a > 1$,
$$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$
I know the definition of limit but I feel that it's not enough to prove this theorem.
NonalcoholicBeer
- 7,211
110
votes
2 answers
$A$ and $B$ disjoint, $A$ compact, and $B$ closed implies there is positive distance between both sets.
Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$.
Proof. Assume the contrary. Let $\alpha_n\in…
Benji
- 5,880
110
votes
8 answers
If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable
I am going over a tutorial in my real analysis course. There is
an proof in which I don't understand some parts of it.
The proof relates to the following proposition:
($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is…
Belgi
- 23,150
110
votes
4 answers
Factorial and exponential dual identities
There are two identities that have a seemingly dual correspondence:
$$e^x = \sum_{n\ge0} {x^n\over n!}$$
and
$$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$
Is there anything to this comparison? (I vaguely remember a generating function/integration…
Mitch
- 8,591
110
votes
2 answers
Connected metric spaces with disjoint open balls
Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls.
Are there any other metric spaces with this…
Hagen von Eitzen
- 374,180
110
votes
23 answers
Complete course of self-study
I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics.
I request people to tell me what topics an…
user37450
110
votes
8 answers
Prove the theorem on analytic geometry in the picture.
I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove?
Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the discussion.
Edit:
Current progress: The theorem…
user122049
- 1,632
110
votes
9 answers
If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?
Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair?
Furthermore, would it still be considered unfair for $50$…
Adam Freymiller
- 1,081
110
votes
6 answers
Why is a geometric progression called so?
Just curious about why geometric progression is called so. Is it related to geometry?
dark32
- 1,401
109
votes
12 answers
Logic puzzle: Which octopus is telling the truth?
King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth.
One day, four servants met.
The blue one says, “Altogether, we have 28…
a1bcdef
- 2,265
109
votes
15 answers
Ways to evaluate $\int \sec \theta \, \mathrm d \theta$
The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a substitution with $u = \sec \theta + \tan…
Mike Spivey
- 55,550
109
votes
6 answers
The Langlands program for beginners
Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program and its geometric analogue?
What are the good books…
ABC
- 1,983
109
votes
4 answers
Projection map being a closed map
Let $\pi: X \times Y \to X$ be the projection map where $Y$ is compact. Prove that $\pi$ is a closed map.
First I would like to see a proof of this claim.
I want to know that here why compactness is necessary or do we have any other weaker…
M.Subramani
- 1,441