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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3answers
34 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 ...
1
vote
2answers
51 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
75 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
56 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
61 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
33 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
26 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
58 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
62 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
124 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
19 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
64 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
36 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
105 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
48 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
61 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
24 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
27 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
81 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
36 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
220 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
65 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
136 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
57 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 ...
0
votes
0answers
23 views

Comparison of books that teach proof techniques

I have to take discrete math and want to learn proof techniques both to get ahead in it as well as open up the possibility of understanding higher math. I've seen several books recommended such as How ...
0
votes
2answers
39 views

Prove that $a+6b+4$ is full square. [on hold]

Let $b=333...33, a=999...99$ where $a$ has $2n$ digit, $b$ has $n$ digit Prove that $a+6b+4$ is full square.
0
votes
0answers
81 views

How do i remove my Guilt? [closed]

When i see a theorem ,which i cannot prove in real analysis ,i think about it but still i couldn't figure it out .Then i look for its solution ,after understanding the proof i feel very guilt that i ...
1
vote
2answers
31 views

GCD Proof Questions

I am preparing for a discrete mathematics exam and am having trouble producing proofs for the following: 1.) Prove that $\gcd (a,c)=1 \Rightarrow \gcd (a,b)= \gcd (a,bc)$ 2.) Prove that $\gcd ...
1
vote
1answer
62 views

Prove that $\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$ has no rational roots.

Show that the following equation has no rational roots. $$\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$$ This is what I've tried: $$\left ( \frac{2\pi x}{x^2+x+1}=\frac{\pi}{6}+2k\pi ...
0
votes
2answers
51 views

How do I show :$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2$? [closed]

How do i show this : $$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2\text{ ?}$$ Thank you for any help
2
votes
3answers
34 views

Prove correctness of simple greedy algorithm to find max

We have $2n$ values $x_1,x_2,x_3,\ldots,x_n$ and $y_1,y_2,y_3,\ldots,y_n$ such that the pair $(x_i,y_i)$ represents the location of a city $i$. Assume there is no straight line that goes through all ...
1
vote
3answers
34 views

How to prove that the cross product of a countable and uncountable set is uncountable?

so my question is, how can you prove that ${\Bbb Z}$ x ${\Bbb R}$ is uncountable? So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm ...
0
votes
1answer
21 views

Proof If a tree is not trivial, then there are at least two pendant vertices?

I have the following Proof but could not understand it Proof. If a tree has $n(≥ 2)$ vertices, then the sum of the degrees is $2(n − 1)$. If every vertex has a $degree ≥ 2$, then the sum will be $≥ ...
3
votes
0answers
46 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
4
votes
1answer
34 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...
1
vote
1answer
20 views

Exlamation about a claim of an existing such cycle in a simple Graph

Suppose the following situation: this is found at (Let G be a graph of minimum degree k > 1. Show that G has a cycle of length at least k+1) Let $P=v_0v_1 \dots v_l$ be a longest path in $G$. ...