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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20
votes
5answers
1k views

Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
0
votes
1answer
23 views

An upper bound to this fraction

The following is an expression I am trying to upper bound by a constant $$I=\frac{x}{1+2y}\leq \ ?$$ The condition that I am using is $$ 2 x < y $$ I have tried the following $$ I = ...
1
vote
1answer
34 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
5
votes
5answers
154 views

Showing uniqueness of integers in base 3

I have recently begun self-studying Number Theory, and am working on proving: Show that every integer $n>0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and ...
0
votes
0answers
40 views

Showing an equality with sequences and sets

Let $a,b$ be two real numbers and $(a_n) \subset \mathbb{Q}$ be a decreasing sequence of rationals such that $a_n \to a $. Also, take a strictly increasing sequence $(b_n) \subset \mathbb{Q} $ such ...
2
votes
2answers
61 views

How to identify an error in a proof?

Right now I'm studying how to find errors in proofs by looking for common mistakes such as circular reasoning, using examples etc. I haven't had too many problems for the most part but I've run into a ...
3
votes
1answer
65 views

Density of a set around $0$ and on $\mathbb{R}$

In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$. Why is that enough to prove that it is dense on $\mathbb{R}$ ?
0
votes
2answers
33 views

how to show that a function is unbounded?

How to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded. I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$
1
vote
2answers
62 views

Proof strategy for basic proofs.

I'm currently in a discrete mathematics course and I'm having quite a bit of trouble with the idea of proofs. From what I understand the one I've been stuck on is also rather simple but to me it's ...
2
votes
3answers
89 views

Showing that a composite number has a small prime divisor?

At the moment I'm working on proving some statements and I've run into one that I can't seem to wrap my head around. It goes like this: For $n \in \mathbb{Z}^+$, we define $\sqrt{n}$ as the real ...
6
votes
2answers
152 views

Can you formalize the proof that $(1 + 2 + \cdots n)^2 = 1^3 + 2^3 + \cdots + n^3$ given here?

This website gives the following proof without words for the identity $(1 + 2 + \cdots n)^2 = 1^3 + 2^3 + \cdots + n^3$. I find it interesting but have trouble seeing the proof behind it. Could ...
2
votes
4answers
82 views

Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$

Show that $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$ Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$. ...
3
votes
4answers
64 views

Proof of divisibility using modular arithmetic: $5\mid 6^n - 5n + 4$

Prove that: $$6^n - 5n + 4 \space \text{is divisible by 5 for} \space n\ge1$$ Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using ...
5
votes
6answers
130 views

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$ I seem to struggle with this simple proof. All I managed to find is that ac+bd=-4 (which might not even be correct).
0
votes
1answer
44 views

Inference rule for Non-Empty Domains

I am currently experimenting with logic frameworks. I am basically using something along dependent types as in "Proof-assistants using Dependent Type Systems" by Henk Barendregt and Herman Geuvers. ...
17
votes
1answer
188 views

How are long proofs “planned”?

I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't ...
1
vote
1answer
94 views

Proof on integrating factors that are a function of one variable

Here's another in my long series of study questions on ODEs (and more to come). I want to prove the following: The (non-exact) equation: $$P(x,y)\,dx + Q(x,y)\,dy = 0$$ Admits an integrating ...
3
votes
3answers
87 views

Pythagorean type diophantine equation.

How to find all solutions to $$ a^2+b^2+c^2+d^2=e^2+2$$ where all variables $a$ to $e$ are positive integers and $e^2 \equiv 1 \mod 8$ I tried using parameterization similar to ...
0
votes
0answers
34 views

How can I prove these generalizations of Weierstrass?

I have thought of the following generalizations of Weierstrass' theorem, but I'm not sure about how to prove them. (1) Let $f:[a, +\infty[ \to \mathbb{R}$ continuous. Then, ...
0
votes
0answers
28 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
2
votes
3answers
73 views

Proving $ C = D $ from $ A \triangle C = A \triangle D$ [duplicate]

I have been working on one part of the proof where my aim is to show that $C = D$. I have been able to prove that $ A \triangle C = A \triangle D$. Is it reasonable to conclude $ C = D $ from $ A ...
3
votes
3answers
60 views

$f: X \mapsto X$ is a function with $f^n = id_X$ for a $n \geq 1$. Prove that $f$ is a bijection.

The problem above is easy to see with $n = 1$, because then every element of $X$ maps to itself and the function $f$ is obviously bijective. By $n = 2$ we have for every $x \in X$ one $y \in X | f(x) ...
1
vote
2answers
57 views

Proving sum of digits of $111111…^2$ is square of sum of digits of $11111…$

How do you prove that the sum of the digits of the square of a number comprised solely of ones is the square of the sum of the digits of that number? For instance, the sum of digits of $111^2$ is 9, ...
2
votes
0answers
63 views

Prove that: if $T$ is an irreducible linear operator then $T$ is cyclic

Let $T:V\to V$ be a linear operator on a finite dimensional vector space $V$. I need to prove that: If $T$ is irreducible then $T$ is cyclic My definitions are: $T$ is an irreducible linear ...
0
votes
0answers
43 views

Spivak proof for Polynomial existence of a root.

Spivak is proving that a odd function $f(x)$ has atleast one root, I almost understand, I just need a little help. The part I dont understand is $(*)$?? I see why he does so that the inequality ...
5
votes
0answers
57 views
1
vote
0answers
31 views

Find all $x \in \Bbb Z \times \Bbb Z_{2}$ s.t. $\Bbb Z \times \Bbb Z_{2} = \langle x \rangle \times \langle0,1\rangle$.

Find all $x \in \Bbb Z \times \Bbb Z_{2}$ such that $\Bbb Z \times \Bbb Z_{2} = \langle x \rangle \times \langle0,1\rangle$. I know the answer is: $\langle1,1\rangle$,$\langle ...
0
votes
1answer
60 views

Use the intermediate value theorem to show that this equation has $n-1$ solutions

I am having this equation: $$ \frac{1}{x-a_1} + \frac{1}{x-a_2} + \cdots + \frac{1}{x-a_n}=0 $$ where $a_1 < a_2 < \cdots < a_n$ are real numbers. Now I want to prove with the intermediate ...
0
votes
1answer
64 views

For $B=\{|x-y|: x,y\in A\}$, show that $\sup B = \sup A - \inf A$ and find $\inf B$

I'm trying to solve this exercise: Let $A$ be a non-empty set $\mathbb R$-bounded. Let $B=\{|x-y|: x,y\in A\}$. Prove that $B$ has a least upper-bound and a greatest lower-bound. ...
1
vote
2answers
42 views

Set theory proof problem about bounds [duplicate]

Suppose $A \ne \emptyset$ is bounded below. Let $-A$ denote the set of all $-x$ for $x$ in $A$. Prove that $-A \ne \emptyset$ that $-A$ is bounded above, and that $-\sup(-A)$ is the greatest lower ...
6
votes
1answer
220 views

Prime Number Sieve using LCM Function

How to prove following conjecture ? Definition : Let $b_n=b_{n-2}+\operatorname{lcm}(n-1 , b_{n-2})$ with $b_1=2$ , $b_2=2$ and $n>2$ . Let $a_n=b_{n+2}/b_n-1$ Conjecture : Every term of ...
1
vote
3answers
102 views

How to prove the intermediate value theorem when $f : \mathbb{R} \to \mathbb{R}$?

Here's the theorem: "The intermediate value theorem states the following: Consider an interval $I = [a, b]$ in the real numbers $ℝ$ and a continuous function $f : I → ℝ$. Then, If $u$ is a number ...
0
votes
0answers
53 views

Show that $\Psi_{t*}\mathbb{X}-\Phi_{t*}\mathbb{X}=\mathbb{X}-\mathbb{Y} $

Let $\mathbb{X},\mathbb{Y}$ be vector fields and let $\Phi_t$ denote the flow of $\mathbb{X}$. Given that $\displaystyle \frac{\partial}{\partial t}\Phi_{t*}\mathbb{Y} ...
4
votes
1answer
37 views

How to prove that an existence statement cannot be constructive

Given the well known spaces of sequences: $$ l_\infty =\{(x_n), n\in \mathbb{N}, x_n \in \mathbb{R} : \sup_n |x_n|<\infty\} $$ $$ l_1= \{(x_n), n\in \mathbb{N}, x_n \in \mathbb{R} : \sum_n ...
3
votes
1answer
78 views

Putnam Problem A-1 2008 3 variable function

I looked at a Putnam problem from 2008, here it is: Putnam Link " Let $f : R^2 → R$ be a function such that $f(x, y)+ f(y,z)+ f(z, x) = 0$ for all real numbers $x, y, z$. Prove that there exists a ...
5
votes
4answers
59 views

Prove that $|x|\leq c\iff -c\leq x\leq c$

I want to show $|x|\le c$ is equivalent to $-c\le x\le c$. But I've taken this for granted so long I'm not actually sure where to start. Can someone give me some hints (not the full solution).
2
votes
2answers
52 views

What to conclude from $ x \in (A \setminus B \cap B \setminus C)$

I have been working on one of the proof of logical statement and one part of it is like this: $ x \in (A \triangle B) \cap (B \triangle C)$ $ x \in (A \setminus B\cup B \setminus A) \cap (B ...
0
votes
1answer
12 views

The longest increasing subsequence of a reversed sequence and a negated sequence

Let's say you have a sequence $A$, for example $1, 5, 2, 3, 6$. You take the reversed sequence: $6, 3, 2, 5, 1$ and the negated sequence: $-1, -5, -2, -3, -6$ and find the length of the longest ...
0
votes
0answers
27 views

Cardinality of any two sets is comparable: proof using Zorn's lemma [duplicate]

Definition 1. Let $A$ and $B$ be sets. $A$ has lesser or equal cardinality to $B$ iff exists a injection from $A$ to $B$. I tried to prove: Let $A$ and $B$ be two non-empty sets such that ...
2
votes
1answer
93 views

Reasons for convergence

I am interested to know if anyone can see the reasoning behind the convergence $$\int_{\Omega}c(u_{k},\nabla u_{k})(u_{k}-u)dx \rightarrow 0$$ in equation (2.82), page 50 in the following book ...
1
vote
2answers
56 views

Proving set equality?

I was looking online for some examples to aid my understanding of set equality and I'm confused on this question. $A = \{x\mid 0 ≤ x ≤ 50 \quad\text{and}\quad x=4n^2 +11, \quad n\in N\}$. $B = ...
1
vote
1answer
33 views

Another form of the Sandwich theorem (for derivatives in dimension $1$)

Here is the theorem : "Let $I\subseteq \mathbb{R}$ an interval which contains $a\in \mathbb{R}$. Let $M$ and $m$ two functions defined on $I$, differentiable at $a$ and $f$ a function defined on $I$ ...
1
vote
0answers
138 views

base change of exterior powers

Let $n\geq 0$ be an integer, $R\to R'$ a ring homomorphism, and $M$ an $R$-module. Then the following holds: $$\bigl(\bigwedge^n_R M\bigr)\otimes_r R' \cong \bigwedge_{R'}^n\, (M\otimes_r R').$$ I ...
0
votes
1answer
52 views

Proof of the Curtis-Hedlund Theorem: Why is there a function $\mu\colon A^S\to A$ such that $\tau(x)(1_G)=\mu(x_{|S})$ for all $x\in A^G$?

Here is the Curtis-Hedlund Theorem and its proof [the sets $V(\cdot,\cdot)$ used in this proof are explained below.]: My problem is I am not sure that I have understand that correctly. So I ...
2
votes
1answer
61 views

Question on the Squeeze theorem

In this theorem we consider the functions $f$, $g$ and $h$ which are defined on $\bar {\mathbb{R}}$ except possibly at $a \in \bar {\mathbb{R}}$ but could we have the limit in $a$ of these three ...
0
votes
1answer
31 views

Which of the following conditions must necessarily be true?

Suppose that $\{A, B\}$ is a set of mutually exhaustive conditions, and that $\{C, D\}$ is another set of mutually exhaustive conditions. If the following implications are true: $$A \Longrightarrow ...
3
votes
1answer
46 views

Motivation for constructing $F$ s.t. $\ker(\text{curl}) \subset \text{Im}(\text{grad})$, $\ker(\text{div}) \subset \text{Im}(\text{curl})$

In 'from calculus to cohomology', we consider the space $V$ of smooth functions $U \to R^3$, with $U \subset R^3$ star-shaped (i.e. convex), and for cohomology reasons (showing $H^1(U)=H^2(U)=0$) we ...
2
votes
2answers
56 views

Upperbound this difference between two log expressions

I have the difference between the following log expressions and I am trying to bound the difference, $$F= \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) - \log \left(1+ ...
2
votes
1answer
87 views

Inequality.such as Nesbitt

Let $a,b,c >0 $ , prove that: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} \leq \dfrac{a^2}{b^2+c^2}+\dfrac{b^2}{c^2+a^2}+\dfrac{c^2}{a^2+b^2}$$
2
votes
1answer
31 views

Shuffling cards and laying them out in order

The numbers from 1 to 50 are printed on cards. The cards are shuffled and then laid out face up in 5 rows of 10 cards each. The cards in each row are rearranged to make them increase from left ...