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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0
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4answers
39 views

Logic, writing proof

i)Suppose that $x$ and $y$ are real numbers. Prove that if $x\neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$ ii)Suppose that $x$ and $y$ are real numbers. Prove that if $x^2y=2x+y$, ...
1
vote
2answers
18 views

Proof that if a simple Graph contains at most two nodes with odd degree then it has a Euler walk

My proof would be start as the following : In general if there are two node at most, then one node used to start walking and the other to end. A) If we start from odd one, this means we have two ...
0
votes
0answers
34 views

Principle of infinite descent

I'm trying to prove the following: "It is not possible to have a sequence of natural numbers which is in infinite descent. (Hint: assume for sake of contradiction that you can find a sequence of ...
3
votes
0answers
48 views

Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$.

I want to prove the following statement: Let $A \in \Bbb R^{n\times n}$ with eigenvalues $\lambda$ and eigenvectors $v$. Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$. ...
1
vote
2answers
49 views

Proving associativity in Algebra

How to proof that a specially defined Transitive Join for the relations $R \subseteq A$ x $B$ und $S \subseteq B$ x $C$ is associative? The join is defined as: $R \Join S =_{def} \{(a,c)| $ there is ...
5
votes
2answers
315 views

proof by contradiction puzzle

Consider the following game between two players: • There is an initially rectangular grid of cookies. • The cookie in the upper left corner is poisoned. • The players take turns. On a player’s ...
2
votes
1answer
60 views

prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
2
votes
2answers
50 views

Forming natural numbers with positive consecutive integers

I'm trying to prove that any natural number N can be formed by adding at least two positive consecutive integers except for powers of 2. For example, using $\,N = 3$, $N = 1 + 2$. When experimenting ...
1
vote
1answer
29 views

Proof Strategies for Convergent Sequences

I am struggling to understand how to choose epsilons during proofs for convergent sequences. It seems that many proofs just state the epsilon to choose without any motivation? How should I go about in ...
0
votes
0answers
44 views

How to prove it and how to solve it

Tomorrow I will begin my studies, real analysis, however I have some difficulties in making statements so I thought before starting the study in real analysis, learn how to do demonstrations properly. ...
0
votes
1answer
54 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
1
vote
1answer
22 views

Proof by contradiction - Predicates and quantifiers

Consider statement, For all integers, b,c,d, if x is a rational number such that $x^2+bx+c=d$, than x is an integer. a) express above statment in the form, $Q_1 b,c,d\in U_1 ( Q_2 x\in ...
0
votes
3answers
38 views

complete the proof for this statement

$$\forall x \in \mathbb{R}, x \neq 0 \implies \frac{1}{x^2\:+3}\:<\:\frac{4}{5}\: $$ I thought of doing the contrapositive but not sure what to do next. $$ \frac{1}{x^{2\:}+3}\:\ge ...
2
votes
2answers
54 views

Epsilon-Delta proof of $\lim_{x\to 2} x^2=4$

I have seen and understand the delta-epsilon proof of the limit of $x^2$ for $x\to2$, such as explained here: https://www.youtube.com/watch?v=gLpQgWWXgMM Now I am wondering, is there also another ...
4
votes
4answers
84 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
1
vote
3answers
66 views

Some questions about proofs of irrational numbers

I have some questions about some things I want to clarify in regard to basic questions that ask to show that roots are irrational, for example $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. To me, I think ...
3
votes
1answer
63 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
0
votes
1answer
35 views

Proving an Iff Statement

Suppose we had a function defined over the complex numbers: $ f(x)= \begin{cases} 1&\text{if } x\in\mathbb{R}\\ 0&\text{if } x\not\in\mathbb{R} \end{cases} $ And we are asked to prove that ...
2
votes
1answer
27 views

Commutator ideal of reductive Lie algebra

I'm working through Fulton and Harris's book on Representation theory, and I've just done the exercise where I had to show: If $\mathfrak{g}$ is a reductive Lie algebra (defined as $Z(\mathfrak{g}) = ...
3
votes
3answers
61 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [on hold]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
1
vote
0answers
30 views

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. [duplicate]

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. I have a problem proving the direction according to which $A$ is compact. First direction I said: If $A$ is ...
1
vote
3answers
65 views

Implies in a truth table, unclear. [duplicate]

In my textbook, we have the following truth table: $P$ true and $Q$ true means that "$P \implies Q$" is true. $P$ true and $Q$ false means that "$P \implies Q$" is false. $P$ false and $Q$ true ...
1
vote
1answer
16 views

Showing an outerplaner graph has less than $2n-3$ edges

An outerplanar graph is a connected plane graph that can be drawn in such a way that all it's vertices are on the outer face. I want to show that for every $G$ outerplaner graph with $n$ vertices and ...
1
vote
6answers
133 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
0
votes
0answers
22 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
1
vote
0answers
40 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
-1
votes
2answers
67 views

Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
1
vote
1answer
37 views

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$.

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$. I have an attempt. I know, or at least think, that it is correct ideally, but I don't know how to make ...
1
vote
1answer
18 views

Confusion regarding differences between strong induction and simple induction

I don't know how to prove that any proof by induction is also proof by strong induction nor any proof by strong induction can be converted into a proof by simple induction? An example would be useful ...
4
votes
2answers
106 views

Function that is continuous and its differential is continuous

Let $ f: \mathbb{R} \rightarrow \mathbb{R}$ . Show that $f$ is continuously differentiable if and only if, for every $x \in \mathbb{R}$ there exists a $l \in \mathbb{R}$ with the property that ...
6
votes
2answers
49 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges ...
1
vote
2answers
62 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
1
vote
2answers
27 views

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$.

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$. I am not exactly sure what to do to show that $H=V$. So far I have reasoned that since $H$ and $V$ ...
0
votes
1answer
18 views

How do we prove a method is optimal?

This is a very simple question, infact it's so simple that I have no idea how to solve it. We have an ordered list of $n$ words. The length of the $i$'th word is $W_i$. Our goal is to write all the ...
3
votes
1answer
29 views

$\varepsilon$ - closeness property

I'm studying Analysis from Terence Tao's book 'Analysis 1' and in an exercise he asks to prove seven properties regarding the notion of '$\varepsilon$ - closeness', which is defined as follows: ...
1
vote
3answers
89 views

Prove that there are infinity many tautologies.

For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by ...
0
votes
0answers
10 views

Theorem implication/equivalence transitiveness in demonstrations

Suppose having three theorems $A, B, C$ that it's necessary to show being equivalent and having the following hypothesis: We know that $A \Leftrightarrow B$ and $B \Leftrightarrow C$. It would ...
-3
votes
1answer
37 views

Is this mathematical statement? [on hold]

$\{\text{integers $n$ such that $n$ is even}\}$ It can be true/false so does that mean it's proposition/mathematical statement?
2
votes
4answers
221 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
0
votes
1answer
69 views

Can this expression be made true ? 2 _ _ _ _ = 2015

Make this expression true: 2 _ _ _ _= 2015 The underscores must be replaced by any 2 of of the operational symbols +, - , x, / (divide). And any 2 of the digits 0,1,2..9. So, you basically need 2 ...
4
votes
2answers
137 views

Proving linearity of a function in two steps

When a function is additive, $$\color{blue}{f(x+y)=f(x)+f(y)},$$ you can extend the property to the product by an integer $$f(nx)=f(x+x+\cdots x)=f(x)+f(x)+\cdots f(x)=nf(x),$$ then to the product ...
3
votes
1answer
60 views

Another Evaluation of the Ramsey number $\mathcal{R}(3,3,3)$

The problem Show that $\mathcal{R}(3,3,3)=17$ The story behind the problem and some notation It was first proven by Greenwood and Gleason in 1955 in their paper Combinatorial relations and ...
2
votes
3answers
32 views

Elementary set theory notation verification

Reading Velleman's "How To Prove It" I came across the following expression: $$ x \in\bigcup\{\mathscr P(A)\mid A\in \mathcal F\} $$ such that $\mathcal F$ is a family of sets, $A$ is a set, and ...
2
votes
1answer
24 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
0
votes
1answer
27 views

Strictly monotonic increasing function with a closed domain and range

Let $a,b,c,d \in \mathbb{R}$ with $a<b$, $I = [a,b]$. Let $f: I \rightarrow \mathbb{R}$ be a monotonic, strictly increasing function. Also $c<d$ and $f([a,b]) =[c,d]$ a) Proof that $f$ is ...
1
vote
2answers
86 views

if integral $f(x)\cdot g(x)=0$ mean that $f(x)=0$?

The question: If $f(x)$ is a continuous function, such that for every continuous function $g(x)$ defined over $[a,b]$ $$\int_a^b f(x)\cdot g(x)\,dx =0$$ does it mean that $f\equiv 0$? The ...
0
votes
1answer
30 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
0
votes
3answers
29 views

Proof:Taylor expansion of inverse trigonometric functions

I find it quite difficult to remember the Taylor expansion of inverse trigonometric functions.Actually in school we have been just taught the series (for finding limits in calculus without teaching us ...
7
votes
1answer
44 views

Basic absolute value property

Hello all I am wondering if anyone has the correct proof that I should use for Spivak calculus ( chapter 1, question 12 ) that says $$|xy|=|x| \cdot |y|$$ from past times I know it is true , but I ...
0
votes
0answers
85 views

Proof, if $f(x) = g(x)$ then $\int f(x)=\int g(x)$.

Let $f(x)$ an $g(x)$ be integrable functions over $[a,b]$ and let $\alpha$ be a point of $[a,b]$ if $f(x) = g(x)$ for all $x\neq \alpha$, then $$\int_{[a,b]}f(x)dx=\int_{[a,b]}g(x)dx.$$ So is it okay ...