For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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3answers
76 views

Can someone check the solution to this recurrence relation?

Here's the recurrence relation: $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ Here's the solution:Write: $$ a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = ...
0
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2answers
58 views

Finding this solution to a recurrence relation

So, I know that the recurrence relation $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ has the solution of $a_n = -4(2^n) - n^2 / 4 - 5n / 2 + 1/8 + (39/8)(3^n)$. I just ...
1
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3answers
59 views

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,…$

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,...$ This is required to prove that the series $1-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{4}$ is divergent, but I ...
2
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1answer
472 views

Limit of Binomial distribution

In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
0
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1answer
28 views

How to show all solutions for a particular recurrence solution

I've found that the recurrence relation $a_n = 4_{an−1} − 4a_{n−2} + (n + 1)2^n$ has the solution of $an = 2^n(p_0 + p_1n + n^2 + n^3/6)$. I'm just trying to understand the steps necessary to solve ...
1
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1answer
70 views

Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
1
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2answers
70 views

Find the recurrence solution of this relation

How would we find the solution of the recurrence relation: $a_n = 2a_{n−1} + 3 · 2^n$ ? After trying it, I've found it to be $a_n = 2^{n-1} (c_1 + 6n)$ Not sure if this is right.. Thanks!
1
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1answer
44 views

On the Dimensionality of Space: An Elementary Analysis

The below theorem I am to prove. Perhaps you have a critique... Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
2
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4answers
174 views

Orthogonal matrices proof

Let $\upsilon _{n} $ be the set of all $n \times n$ orthogonal matrices(for all fixed $n$). Show that $\upsilon _{n} $ is not a subspace of $\ M _{n\times n} $. Thank you ! Additional: Suppose $A ...
0
votes
2answers
77 views

Are the lengths from this recursive construction a geometric sequence?

In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
1
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1answer
60 views

Prove that the Iwata function is Submodular

The Submodularity property for $f: 2^V \rightarrow \mathbb{R}$ is defined as: $f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ where $X, Y \subseteq V$ While the Iwata function is defined as: ...
2
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2answers
138 views

Finding the solution to this specific recurrence relation

What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$, $a_1 = 10$, and $a_2 = 32$ I can find it for a specific value of (n), but not for just a general solution. Thanks!
3
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0answers
35 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
1
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1answer
63 views

Proof Technique and Factorials

I need to prove that $\;n!+m$ is divisible by $m$ for all integers $n \ge 2$ and $1 \le m \le n$.
1
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0answers
99 views

Proving a specific recurrence relation theorem

I'm trying to come up with a proof for this theorem: Let $c_1$ and $c_2$ be real numbers with $c_2 != 0$. Suppose that $r^2 - c_1 r - c_2 = 0$ has only one root, $r_0$. A sequence $\{a_n\}$ is a ...
1
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2answers
547 views

Finding a solution to a recurrence relation

Find the solution to $$a_n = 5a_{n−2} − 4a_{n−4}$$ with $$a_0 = 3$$ $$a_1 = 2$$ $$a_2 = 6$$ $$a_3 = 8$$ My answer: Observe that the degree of recurrence is 4. Hence, the characteristic equation is: ...
3
votes
1answer
242 views

Finding a Linear Recurrence Relation

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. ...
1
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0answers
92 views

Is this theorem proof correct?

I'm trying to prove this theorem: Let $c_1$ and $c_2$ be real numbers with $c_2 \ne 0$. Suppose that $r^2 − c_1r − c_2 = 0$ has only one root $r_0$. A sequence $\{a_n\}$ is a solution of the ...
1
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2answers
117 views

Strategies to prove inequalities with interval notation

How to prove a inequalities with interval notation, for example: Find minimum of $a^3+b^3+c^3$ with $a,b,c \in [-1;\infty), a^2+b^2+c^2=9$
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2answers
50 views

Help with real number proof strategy

Let $x, y, z \in \mathbb R$. Prove that $\displaystyle \frac{|x-y|}{1+|x-y|} \leq \frac{|x-z|}{1+|x-z|} + \frac{|z-y|}{1+|z-y|} \; $ Help on how to start with this one? Real lost.
1
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0answers
97 views

A seemingly complicated proof.

Let $$ f(x) = (ax)^{\big(b\sin x\big)} $$ $$ g(x) = (x - \pi/2)(\sin x)^{\cfrac 1 {(x-\pi/2)}} $$ Also, let $ h(x) = g(x) - f(x) $ be defined in $(\pi/2, \pi) $ If $h(x)$ is an increasing function ...
1
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1answer
63 views

Inverse implies surjection and follow-your-nose proofs

(I'm posting this question with my own answer, to show a nice calculational proof for one of the examples in Luke Palmer's blog post Follow Your Nose Proofs.) In what follows, $A$ and $B$ are sets, ...
0
votes
1answer
143 views

Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
4
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2answers
83 views

If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.

I'm reading through How to Prove It by Velleman and I'm having trouble with this exercise in the section about Existence and Uniqueness proofs. Here is the exercise: Suppose $A$ is a set and for ...
2
votes
2answers
93 views

Proving $r_0+r_1a+r_2a^2+\cdots+r_{k-1}a^{k-1} < a^k$ by INDUCTION.

Let $a$ be a natural number $>1$. For all integers $r_0, r_1, \dots, r_{n-1}$ with $0\leq r_{j} < a$, then \begin{eqnarray} r_0+r_1a+r_2a^2+\cdots+r_{n-1}a^{n-1} < a^n. \end{eqnarray} ...
1
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2answers
67 views

Isomorphism of Posets

Let $(X,\le),(Y,\le)$ be posets. $\text{Iso}(X,Y)$ denotes the set of isotones from $X$ to $Y$. $f:X\to Y$ is an isotone if $x_1\le x_2 \implies f(x_1)\le f(x_2)$. $(A,\le),(B\le),(C,\le)$ are ...
0
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1answer
107 views

What is the official proof (if there is any) for the area of a circle of radius 'r'?

What is the official proof (if there is any) for the area of a circle of radius 'r' ? I remember in my school days they simply told that area of a circle of radius 'r' is $\pi*r^{2}$. The teacher ...
6
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0answers
290 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
2
votes
0answers
192 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
6
votes
0answers
370 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
1
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0answers
75 views

What are some general approaches to proving smoothness?

What are some general strategies for proving that a given function $f(x)$ is smooth (continuous in all orders of derivatives)? What properties of a function are needed to carry out a proof? $f(x)$ ...
1
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2answers
164 views

Prove the following ceiling and floor identities?

Could someone help me prove these identities? I'm completely lost: $$\begin{align*} &(1)\quad \left\lceil \frac{\left\lceil \frac{x}{a} \right\rceil} {b}\right\rceil = \left\lceil ...
0
votes
1answer
43 views

Euclidean space problem

In three-dimensional space, is it true that if you take line $a$ of a plane and line $b$ of the plane perpendicular to the first one, then the angle between line $a$ and $b$ (at which they intersect) ...
1
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0answers
180 views

Question about proof of bounded real lemma

My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure? The $H_\infty$ performance is defined as: \begin{align} \parallel ...
4
votes
3answers
260 views

Proving or Disproving the Sum in a Set

I am doing review questions for an exam and I am completely stumped on this particular question: Let A = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}. Prove or disprove that if I select 10 distinct ...
2
votes
1answer
29 views

Well-founded part of a graph

Let (A,R) be a graph. Define by transfinite recursion: $ W_{0}=\emptyset \\ W_{\alpha+1}=\{a \in A : ext_{R}(a) \subseteq W_{\alpha +1}\} \\ W_{\alpha}=\cup_{\beta < \alpha} W_{\beta} \text{if ...
1
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1answer
233 views

Finding the probability that X will be successful if its success is predicted

Consider an electronics company is planning to introduce a new camera phone. The company commissions a marketing report for each newproduct that predicts either the success or the failure of the ...
3
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1answer
50 views

Binomial coefficient help?

I'm studying for my exams and would appreciate any help with binomial coefficients. I think I got the idea but having trouble with a specific one: Q) If a there are 11 dogs and 9 cats: a) How many 7 ...
1
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1answer
89 views

Prove that the set $\mathbb{R^{n}}$ has the the same size as $\mathbb{R}$.

I would very much appreciate if anyone would be so kind as to review whether or not the following attempt is successful or not. Theorem: The set $\mathbb{R^{n}}$ has the the same size as ...
2
votes
2answers
76 views

is this argument true?

i had a puzzle and used a logical argument to show a point but some says that my argument is wrong but i can't understand the reason they provide ! the puzzles says , Given four cards laid out on a ...
1
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3answers
258 views

For all sets $A$ and $B$, if $A^c ⊆ B$ then $A ∪ B = U$

For all sets $A$ and $B$, if $\;A^c ⊆ B$ then $A ∪ B = U$ I am having difficulty starting to disprove an alleged set property through the use of a counterexample or if it is true then try to ...
1
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3answers
84 views

For all sets $A$ and B, if $B ⊆ A^c$ then $A ∩ B = ∅$

I made a Venn Diagram so I know that this is true. Now I just need some help on getting the proof right. For all sets $A$ and B, if $B ⊆ A^c$ then $A ∩ B = ∅$ I have started the proof: Suppose $A$ ...
1
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2answers
59 views

Prove that the sequence is bounded. [closed]

Define what it means for $x_n$ to be bounded. Prove that a sequence $$x_n=\frac{2n+11}{3n+14}$$ is bounded.
5
votes
3answers
222 views

Set of all triangles with two equal edges inscribed in a circle.

Let $\Delta$ be the set of all triangles with two equal edges inscribed in a circle of radius $R$. So, how do I show that: 1, The equilateral triangle in $\Delta$ is the one maximizing the area. ...
2
votes
2answers
127 views

Prove by contradiction using division algorithm

Let $z$ be a primitive $n$-th root of unity. Prove that for any $k\in\mathbb{Z}$, if $z^k=1$, then $n \mid k$.
-1
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1answer
126 views

Two analysis proofs about infima, suprema, and bounded sets

Can anyone give me a complete proof for the following two questions: Prove that if a set has a minimum (maximum), it must coincide with the infimum (supremum). Show that every non empty finite set ...
3
votes
2answers
64 views

How to move from a right semigroup action to a left semigroup action?

Let $S$ be a semigroup and $X$ any set. Define a left action of $S$ on $X$ to be a map $\sigma: S \times X \rightarrow X$ with the property that $(st)x = s(tx)$, where we define $gx = \sigma(g,x)$ ...
3
votes
1answer
117 views

Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$

Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
5
votes
6answers
2k views

Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
2
votes
1answer
501 views

How to show a graph is not Hamiltonian?

Suppose you are given a graph $G$ with the properties that $G$ is 3-regular, $v_G = 10$ where $v_G$ is the number of vertices in $G$, and girth$(G) \geq 5$. How can you tell that $G$ is not ...