Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?

Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$. Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$...
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Trying to prove a proposition about the nth order derivative of a polynomial by induction - is this correct?

Recently, I decided to try and create a formula for the $n$th order derivative of a polynomial, and I believe I succeeded! I tried to do a proof by induction to confirm this for myself, but since I ...
cdog's user avatar
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Is there a Sokoban level with such conditions

First of all, let me explain what Sokoban is. It is a logic game created in Japan and it literally means "warehouse keeper". It is a type of transport puzzle, in which the player pushes boxes or ...
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11 votes
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Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
precarious's user avatar
9 votes
2 answers
278 views

Controlled natural language for mathematics

I am a French student very inspired by Bourbaki's but I can no longer stand to write approximate proofs. I was wondering if there was a language between formal and natural language that was both non-...
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292 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all positive integers $m$ such that the ratios $$ \frac{2(5^m+5)}{3^m+1}\quad\text{and}\quad \frac{9^m+1}{5^m+5}$$ are both integers. Attempt at a solution: If the ratios are both ...
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About Theorem 3.4 Hartshorne: detailed proof.

I propose a detailed version of part of the proof of Theorem 3.14 from Hartshorne's book Algebraic Geometry. The questions are inserted from time to time within the proof. Thanks for your patience. ...
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Degree of hypersurfaces in Grassmannians

In the book Discriminants, Resultants, and Multidimensional Determinants of Andrei Zelevinsky and Izrail' Moiseevič Gel'fand, the authors give the following definition of degree of a hypersurface in a ...
Vincenzo Zaccaro's user avatar
8 votes
1 answer
198 views

Can a product of a number and its reverse consist of only $1$'s?

Problem: Let $n \gt 1$. If you write the digits of $n$ in reverse, then multiply by original $n$, is it possible for the product to consist only of $1$'s? This came from a competition I recently ...
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Differential geometry: non-uniqueness of the closest point on a curve

I have looked for theorems about the closest points, but I could not find such a theorem I need to establish some other claim. The main question is expressed as a proposition as follows: Proposition: ...
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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, ...
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Prove that if $\lim_{x\to \infty} f'(x) = 0$, then $\lim_{x\to \infty} f(x+1)-f(x) = 0$

I'm working on this proof and I think I have a sketch but I'm not sure it's rigorous enough. Suppose $f:\Bbb R \to \Bbb R$ is differentiable and that$$\lim_{x\to \infty} f'(x) = 0$$ Prove that $$\lim_{...
Bigtalian's user avatar
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Find extrema of $y=?(x)-x$ with the Minkowski Question Mark function

The Goal: is to figure out the global extrema of the Minkowski Question Mark function $?(x)$. Here is the graph of: $$?(x)-x:$$ The $y$ value of the global maximum was found by systematically ...
Тyma Gaidash's user avatar
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Prove there is no smallest positive irrational number.

Proof: Assume by way of contradiction that there is a smallest positive irrational number $x$ where $x\in\mathbb{R-Q}$. Consider $0<x/2<x$, because $x$ is irrational $x/2$ is also irrational. I ...
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Show that $\forall n, a_n$ is an integer that is multiple of $n$.

Consider the sequence $\{a_n\}$ defined by $\;a_1=1,\; a_2=2,\; a_3=24\;$ and $$a_n=\frac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}},\; \forall n\geq 4.$$ Show that $\;\forall n,\; a_n\;$ ...
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Proof that a factorization domain is a unique factorization domain if and only if every irreducible element is prime.

Today in algebra class my professor proved, among other things, that a factorization domain is a unique factorization domain if and only if every irreducible element is prime. I had a hard time ...
isekaijin's user avatar
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6 votes
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Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
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Derivative of a composition of function - nice proof

Let's consider the well known "fake" proof below for the derivative of the composition of functions: Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let $f:E\...
Hippalectryon's user avatar
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Show that a convergent sequence exists within any uncountable set of reals

Assume $A$ is an uncountable set of reals, show there exists a convergent sequence $a_n$ such that $(\forall n \ a_n\in A) \land (\forall n,m \ n\neq m\implies a_n\neq a_m)$ Please check the validity ...
martinkleins's user avatar
5 votes
0 answers
256 views

Trying to prove the Erdős–Straus conjecture

The Erdős–Straus conjecture states that $\frac{4}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$, for any $n∈\mathbb{N}$ and $n\ge2$. I've done a question on this a while back, and recently I decided ...
Mathemagician314's user avatar
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Is $\ \left\{\ \left(\frac{3}{2}\right)^n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ a dense subset of $\ [0,1]\ $?

Is $\ \left\{\ \left(\frac{3}{2}\right)^n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ a dense subset of $\ [0,1]\ $ ? I know how to prove that if $\ \gamma\ $ is irrational, then sets like $\ \left\{\ \...
Adam Rubinson's user avatar
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Suppose $A$, $B$, and $C$ are sets. Prove that $A\cup C\subseteq B\cup C$ iff $A\setminus C\subseteq B\setminus C$.

Not a duplicate of Prove that $A \cup C \subseteq B \cup C$ iff $A \setminus C \subseteq B \setminus C$ Suppose $A$, $B$, and $C$ are sets. Prove that $A ∪ C ⊆ B ∪ C$ iff $A \setminus C ⊆ B \setminus ...
Khashayar Baghizadeh's user avatar
5 votes
0 answers
202 views

The conjecture about the existence of closed-form inverses of functions

Is my proof draft below already a proof? How can the proof be completed? Definition: A unary complex function is a function from a subset of $\mathbb{C}$ into $\mathbb{C}$. A binary complex function ...
IV_'s user avatar
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An important corollary of Hahn - Banach Theorem

Let $X$ a normed linear space. Denote with $\mathbb{F}=\mathbb{C}\;\text{or}\;\mathbb{R}$ Suppose that (1)$\;M$ is a closed subspace of $X$; (2)$\;x_0\in X\setminus M$; (3)$\;d=\text{dist}(x_0, M)=\...
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In proving that $\sqrt{a}$ is always irrational, $\forall a\in\left\{\Bbb R^+ : 1< a\neq b^2\right\}$... a different way.

I was trying to prove the following statement: $$\sqrt{a}\text{ is always irrational, }\forall a\in\left\{\mathbb{R}^+ : 1<a\neq b^2\right\}.\tag{$b\in\mathbb{Z}$}$$ I know there is at least one ...
Mr Pie's user avatar
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Generalized Taylor derivatives test

I am seeking for a proof of the generalized derivative test to find inflection points, minima and maxima. I am seeking for a proof that I read some time ago but can't find anymore. The thesis was that ...
John's user avatar
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How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
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Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
user153012's user avatar
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5 votes
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Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a function, $a∈...
James Taylor's user avatar
5 votes
0 answers
338 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: $$\sin(x)=x\prod_{i=1}^{\infty}(1-\frac{x^2}{i^...
Vim's user avatar
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5 votes
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The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
Lana Cape's user avatar
5 votes
1 answer
396 views

Show that $f^{-1}(B)=A$

I started yesterday my study of functions. I’m following the book “Proofs and Fundamentals”, by Ethan D. Bloch, and I’m having some trouble in starting myself in formal proofs that involve functions. ...
Air Mike's user avatar
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Exercise 11, Section 6.6 of Hoffman’s Linear Algebra

Let $V$ be a vector space, let $W_1, \ldots, W_k$ be subspaces of $V$, and let $$V_j = W_1 + \cdots + W_{j-1} + W_{j+1} + \cdots + W_k.$$ Suppose that $V = W_1 \oplus \cdots \oplus W_k$. Prove that ...
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4 votes
0 answers
61 views

Examples of nontrivial proofs by cases in which only one case is ever realized

For teaching reasons, I'm looking for examples of proofs that use a nontrivial case breakdown in which only one case is ever realized, and yet it is very hard to prove which case is the "real&...
GMB's user avatar
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4 votes
1 answer
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Prove or disprove that the function is convex .

It seems we have : Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$ Claim : $f''(x)\geq 0$ My attempt : The case $n=1$ is trivial . So I have ...
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Proof Verification: Baby Rudin Ch. 6. Ex. 7a

I want to follow up on my previous question. Based on the comments and responses I got for my previous question, I developed a new proof for Baby Rudin Ch. 6. Ex. 7a. The exercise is: Suppose $f$ is ...
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4 votes
0 answers
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To prove the existence of solution(s) of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$

To prove if a certain equation has a solution, we do not need necessarily solve that equation. Example: Is there any point of the curve $y=\sin^2(x)/x$ between $x=0$ and $x=\pi$ so that the slope of ...
Hussain-Alqatari's user avatar
4 votes
0 answers
233 views

$G$ is a finite group and $a \in G$ s.t. $a$ has exactly $2$ conjugates. Then $G$ contains a non-trivial, proper normal subgroup.

$G$ is a finite group and $a \in G$ s.t. $a$ has exactly $2$ conjugates. Then $G$ contains a non-trivial, proper normal subgroup. This question proved to be more difficult than I had expected, and ...
Sun's user avatar
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4 votes
0 answers
488 views

Problem 8, chapter 1 - Rudin's functional analysis

Trying to solve this problem: Problem 8: a) Suppose $\mathcal{P}$ is a separating family of seminorms on a vector space $X$. Let $\mathcal{Q}$ be the smallest family of seminorms on $X$ that ...
user8469759's user avatar
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4 votes
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Show $\mathrm{det}(M)$ is well defined.

One intuitive way to approach studying the determinant of a given matrix $M$ is to inspire its formal definition in the signed volume of applying the corresponding transformation to the unit $n$-cube. ...
Sam's user avatar
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4 votes
0 answers
556 views

Proof by induction on an uncountable interval

I'm a little bit confused as to whether or not I can use proof by induction on an uncountable interval. For example, I'm trying to prove that $$(x+y)^n \leq x^n + y^n$$ on the interval $0 \leq n \leq ...
bco's user avatar
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0 answers
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Improving clarity and argumentation with hard-to-describe combinatorial proof

I'm doing undergraduate research and the content of my paper depends on the following lemma. I tried something like a combinatorial proof, but it is clearly not rigorous, partly because my argument is ...
actinidia's user avatar
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Prove that $||x+y|| \leq ||x|| +||y||$

From Munkres' Topology, I get this question. A hint suggests us to use a result from a previous subquestion. But it seems that I don't need to use the previous result to prove this. Can someone help ...
user398843's user avatar
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4 votes
0 answers
162 views

Have there been any (interesting) computer-aided proofs that weren't proof-by-exhaustion?

It seems to me like many of the most famous "computer proofs" were done by basically brute-forcing through all of the cases, such as the four color map theorem. Are there any good examples of computer ...
Glu's user avatar
  • 407
4 votes
2 answers
157 views

complex number to a power divisible by 6

I actually have a follow-up question to this post -- given that n is a positive integer such that $z^n = (z+1)^n = 1$, I need to show that n is divisible by 6. I can now show that $z$ and $z+1$ both ...
space's user avatar
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4 votes
0 answers
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How do you prove this implication using its contrapositive?

$∀x∈R,|2x-1|≤5$ and $|2x-1|>3⇒(x^4+7≤7x^2 )$ or $(2x^3≥8x+5)$ This is what I got for the contrapositive: $∀x∈R,(x^4+7>7x^2 )$ and $(2x^3<8x+5)⇒|2x-1|>5$ or $|2x-1|≤3$ Where would I ...
James's user avatar
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0 answers
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Non calculus proof of $SS_1 = T^2$

The equation of a pair of tangents from $(x_1,y_1)$ to the circle $x^2+y^2+2gx+2fy+c=0$ is given by $T^2= SS_1$ where: $S= x^2+y^2+2gx+2fy+c \\ S_1= x_1^2+y_1^2+2gx_1+2fy_1+c \\ T = xx_1+yy_1+g(x+x_1)...
Archer's user avatar
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4 votes
0 answers
194 views

Prove that $\sqrt[m]{r}∉ℚ(\sqrt[n]{q})$

Let $ℚ(\sqrt[n]{q})$ be the field of rational numbers with $\sqrt[n]{q}$ adjoined; $n,q∈ℚ$ I have been trying to prove that $\sqrt[m]{r}∉ℚ(\sqrt[n]{q})$, for different $m,r∈ℚ$. My approach: We write ...
Sam's user avatar
  • 4,676
4 votes
0 answers
234 views

How to make a proof your own?

Often, in the interest of time, We look at proofs done by others. It could be on stack, in a textbook, or shown by a friend. In all cases, you don't get the same amount of mental compression of the ...
Yashmnash's user avatar
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4 votes
0 answers
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How should you refer to other people's result when writing in math?

Here is what I was taught about coming up with my own results in math school: A lemma builds up to a theorem, which implies corollaries. A theorem you are not proud of is a proposition. However,...
Shamisen Expert's user avatar

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