If you already have a proof for some result, but want to ask for a different proof (using different methods).

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3
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0answers
29 views

Shortest proof for showing $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID.

I'm looking for an easy proof for that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID. One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it ...
1
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2answers
20 views

Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces.

I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ...
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0answers
33 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| ...
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0answers
14 views

Proving three asymptotic identities (Murray (1984)'s Exercise 1.1.4)

(Context: I'm self-studying Murray (1984). I learned (and have forgotten quite a lot of) real and complex analysis. I'm willing to relearn and to look up references.) Problem: if $f=O(g)$, show that ...
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1answer
18 views

How to improve my proof and whether or not one condition in the statement is important in writing the proof?

A simple graph $G$ is connected iff for every partition of the vertices into two non-empty sets $X$ and $Y$, there is a vertex $x\in X$ and a vertex $y\in Y$ such that $xy$ is an edge of $G$. My ...
0
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2answers
30 views

$f$ is bijective, show that $h(x)=\left(f(x); g(x)\right) \rm{\ is\ bijective\ } \iff G $ is Singleton

Let E, F and G be three sets ($E\neq 0;F\neq 0,G\neq 0 ) $ Let $h$ defined by : $$\begin{align} h \ \colon\ E & \to F\times G\\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ ...
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2answers
28 views

Is $h(x)=\left(x^2; 1_{[0,\infty)}(x)\right)$ an injective function?

Let h defined by : $$\begin{align} h \ \colon\ \mathbb{R} & \to \mathbb{R}^{2} \\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ and $$\begin{align} f \ \colon\ \mathbb{R} ...
1
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1answer
32 views

Convexity of the field of values

I am looking for alternate proofs for the convexity of the field of values. In Topics in matrix analysis by Horn and Johnson they define the field of values of an $n\times n$ matrix $A$ as $$F(A) = ...
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2answers
25 views

Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph

Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
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2answers
40 views

Constructing topology on $\Bbb{Z}$

Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...
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votes
2answers
37 views

Could anybody please check my proof about connected graph?

I have written a proof of the following statement but not sure whether it is correct or not. Let $G$ be a connected graph and each vertex has even degree. Show that if we remove ANY edge of the graph ...
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2answers
52 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
0
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1answer
35 views

How to fill in the gaps in my proof to make it more convincing?

Let $T$ be a tree with $3$ edges. Let $G$ be a simple graph such that each vertex has degree at least $3$. Show that $G$ has $T$ as a subgraph. This statement is obvious but I am not sure how to ...
1
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1answer
31 views

What is a useful starting idea to think about this simple graph problem?

I would like to prove the statement that there are $2^{\binom{n-1}{2}}$ simple graphs are there with vertex set $\{1,\ldots,n\}$ such that every vertex has even degree. The thing that confuses me is ...
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2answers
69 views

Alternative proof of Euler's result that $641$ divides $2^{32} + 1$

I am looking for an alternative proof of Euler's result that $641$ divides $2^{32} + 1$. I've seen a solution so far, understood the solution, but unfortunately I don't know how to think in order to ...
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1answer
36 views

Convergence of the power of a matrix : a simpler proof?

Here is an exercise that I have been used in a oral exam (around 45 minutes) for undergrad students. Let $A$ a $n\times n$ matrix with real coefficients such that $$ A^T = 3A^2-A-I_n,$$ where $A^T$ ...
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1answer
47 views

Not sure how to prove this statement by contradiction?

There is this a simple looking and intuitive statement but I am not sure how to start approaching this problem. Let $S=\{s_1,s_2,\ldots,s_n\}$, where $s_1,s_2,\ldots,s_n>0$ such that ...
3
votes
1answer
93 views

Subfield of $\mathbb{Q}(\sqrt[n]{a})$

Exercise 14.7.4 from Dummit and Foote Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in \mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$(i.e., $x^n-a$ is irreducible). Let $E$ be any subfield of ...
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1answer
36 views

Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$.

Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$. So my textbook divides the problem into first case $p=1$ and integrates and cases $p<1$ in which it uses integration by ...
0
votes
3answers
59 views

Request for a proof that $\sum\limits_{i=1}^n i^{k+1}=(n+1)\sum\limits_{i=1}^n i^k-\sum\limits_{p=1}^n\sum\limits_{i=1}^p i^k$

Prove $$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$ for every integer $k\ge0$. By principle of induction, $$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$ ...
4
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1answer
30 views

Show $P(\frac{1}{n}\sum_{i=1}^{n}Y_i\geq c)\leq e^{-nd}$ for constants $c$ and $d$

Let $Y_1, Y_2\ldots$ be a sequence of i.id. random variables uniformly distributed on $[0,1]$. Let $c>\frac{1}{2}$. Show that there exists $d>0$ (depends on $c$) such that ...
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0answers
190 views

There exists polynomial $(P_{n})_{n\in\mathbb{N}}$ such that $P_n(0)=1$, and $\lim_{n\rightarrow \infty}P_{n}(z)=0$..

Show that there exists a sequence of polynomial $(P_{n})_{n\in\mathbb{N}}$ such that $P_n(0)=1$ for each $n$, and $\lim_{n\rightarrow \infty}P_{n}(z)=0$ for all $z\in \mathbb{C}\setminus ...
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2answers
28 views

Sum of cardinals of all intersections: elegant alternative proofs?

I once read the following problem: compute $$\sum_{A,B\in\mathcal{P}(\Omega)}\operatorname{card}(A\cap B)$$ where $\Omega$ is a set of cardinal $n>0$ and $\mathcal{P}(\Omega)$ the set of the sets ...
1
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2answers
35 views

Constructing a square from the difference of diagonal and side

I have figured it out how to construct a square given a segment which is the difference of diagonal and side: construct an equal sided right triangle where the leg is the given segment and then add ...
5
votes
1answer
111 views

Proof verification : a very useful theorem (in measure theory)

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
3
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2answers
57 views

Prove that if $2n+1$ and $3n+1$ are both perfect squares then $40|n$.

Prove that if $2n+1$ and $3n+1$ are both perfect squares then $40|n$. First, I took $$2n+1 \equiv x^2 \equiv 0, 1 \pmod 4$$ which showed that $n$ was even. Now, $$3n + 1 \equiv y^2 \equiv 0, 1, ...
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0answers
11 views

Prove that the number of nodes in ORBDD for fn with given order On is 2n +2

I cannot embed image yet, so I have no choice but to include a link here. (Also if anyone can include a link/tutorial/guide to how to display notations, I'd really appreciate it) This image contains ...
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2answers
32 views

Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the ...
0
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1answer
19 views

Multidimensional Cantor diagonal argument for ordering infinite sets [duplicate]

Cantor diagonal argument is a powerful proof technique. It has been used for a lot of proofs. For instance, it has been used to prove that $|\mathbb{N}| < |\mathbb{R}|$. What can we say about the ...
3
votes
3answers
112 views

A purely algebraic proof of $\vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta)$

I have seen a proof of the fact that $$ \vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta) $$ where $\vec{a}$ and $\vec{b}$ are two vectors. The proof relies on the Law of ...
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0answers
19 views

How can I verify the following equality?

$$\int_0^{\infty}\frac{C\exp(-\frac{mx^2}{\Omega})}{\Omega^m}\frac{1}{\sqrt{2\pi}\lambda\Omega}\exp\left(-\frac{(\ln ...
3
votes
1answer
54 views

The well-ordering principle implies Zorn's Lemma

I have read and understood proofs for each implication between $AC$, $ZL$, $WO$ except this one. These proofs need about 10 lines each. Can someone share a neat, hopefully short, proof for $WO\implies ...
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0answers
16 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
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0answers
66 views

Unramified at a point $x \in X$ if and only if $\Omega _{X,x} = 0$

This is Corollary 6.2.3 in Liu's book. Let $f: X \to S$ be a morphism of finite type of locally Noetherian schemes. Then $f$ is unramified at a point $x \in X$ if and only if $\Omega_{X/S, x} = ...
0
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3answers
61 views

Alternative Proof: if $n$ is an integer, prove that $\frac{n ( n^4 - 1)}{5}$ is an integer

I have proven this by the induction method but would like to know if it can be proven using an alternative method.
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1answer
26 views

Probability space for zebras and their number of stripes

On a trip to Africa the researcher Alison notices that zebras with an even amount of stripes have double the probability to be seen than zebras with an odd amount of stripes. Let $E_n$ denote the ...
2
votes
1answer
38 views

Show that finite dimensional subspace is closed

We know that if $V$ is a normed vector space and $W$ is a finite dimensional subspace of $V$, then $W$ is closed. One way to prove this is to show that $W$ is actually complete. Since complete space ...
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1answer
23 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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3answers
23 views

How to determine intervals where $f$ is greater than $g$?

I have two functions, $f(x) = 2x$ and $g(x) = \frac{x^3}{3}$. I solved for $x$ where $f = g$, finding $x = \pm 6^{1/2}$, then solved for $x$ where $f > g$, $x > \pm 6^{1/2}$, and where $f < ...
0
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1answer
34 views

Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$ I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite ...
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2answers
24 views

Show that $\bigcup_{n=1}^\infty A_n= B_1 \backslash \bigcap_{n=1}^\infty B_n$

Let $\{B_n\}$ be a decreasing set $B_1 \supseteq B_2 \supseteq B_3 \supseteq ....$ Define $A_n = B_1 \backslash B_n$ i.e. $A_1 = \varnothing, A_2 = B_1 \backslash B_2$ If we imagine $\{B_n\}$ as a ...
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votes
3answers
67 views

Prove that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is constant

Suppose that the function $f:\mathbb{R}^2\to\mathbb{R}$ has first order partial derivatives and that $$\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0\qquad\text{for all ...
4
votes
3answers
124 views

How do you prove this without using induction?

How do you prove this without using induction $$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n-1}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}$$
24
votes
1answer
279 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
0
votes
1answer
36 views

Geometric Proof of DeMoivre's Formula

I want to know if there were a geometric proof of DeMoivre's formula. My attempt was starting with an easy complex number and observing patterns, then generalizing that pattern. If you start with ...
5
votes
0answers
34 views

More elegant derivation of the shift in median bin occupancy

In answering Median of a multinomial variable, I found to my own surprise through a somewhat tedious calculation that the expected value of the median of the ball counts in $3$ bins into which $n$ ...
2
votes
2answers
195 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca ...
3
votes
3answers
41 views

Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of ...
0
votes
1answer
14 views

A short proof that the set of discontinuities of a regulated function is countable

Does a short proof exist for the following theorem? The set of discontinuities of a regulated function on $[a, b]$ is countable.
0
votes
1answer
28 views

Proving sequence converges

I am trying to prove: Suppose that {an} and {bn} are two sequences such that {an} and {an + bn} converge. Prove that {bn} converges. Here is my first attempt: Proof: Suppose that {$a_n$} and {$b_n$} ...