This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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2answers
30 views

Show that : $\lim\limits_{h\to 0}\frac {h^5} {2h^4} \frac1{\sqrt{h^2 + h^4}} = \lim\limits_{h \to 0}\frac {h^5} {2h^5}$

In my textbook, I found the following step, but I don't understand how the author gets there. $$\lim_{h\to 0} {{\frac {h^5} {2h^4} \over \sqrt{h^2 + h^4}}} = \lim_{h \to 0}\frac {h^5} {2h^5}$$
0
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1answer
24 views

Proof that all isometries can be written under the form $Q(x)+v$

I want to show that all $f \in Isom(\mathbb{R^n})$ can be written as $f(x) = Q(x) +v$ with $Q \in o(n) $ and $v \in \mathbb{R^n}$ This is how the proof goes: Let us set G(x) = f(x) -v We want to ...
3
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0answers
38 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
18
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2answers
1k views

What's going on with this 5-line proof of Fermat's Last Theorem? [duplicate]

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author ...
1
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2answers
43 views

Show, for every connected graph G of order 6 with four independent vertices, that either α(G)=5 or α′(G) ≥ 2. [on hold]

Show, for every connected graph G of order 6 with four independent vertices, that either α(G)=5 or α′(G) ≥ 2. a(G) stands for vertex independent number (max number of vertices such that no two ...
0
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1answer
24 views

Perfectly Normal is hereditary

The definitions I'm working with: $(X, T )$ is called perfectly normal if whenever $C$ and $D$ are disjoint, nonempty, closed subsets of $X$, there exists a continuous function $f : X \rightarrow [...
1
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1answer
41 views

Can't see why particular homotopy is continuous

I'm checking the group laws for the fundamental group of $(X,x_{0})$: in particular I'm trying to show that $\gamma \simeq \gamma \cdot e$ , where $\gamma$ is a loop based at $x_{0}$, $e$ is the ...
1
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1answer
35 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
1
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1answer
26 views

Why does MLE work for continuous distributions?

In the attachment below you can see the definition of the likelihood function. Likelihood 1) Whilst the explanation of why the whole max likelihood method is viable for discrete distributions is ...
0
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1answer
30 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
0
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1answer
20 views

Prove if $x_1,…,x_n$ are natural numbers with $n\geq2$ then $x_1x_2…x_n$ is odd iff $x_i$ is odd for all $i$, $1\leq i\leq n$

I am not sure if Im on the right track here but if any one could help out I would greatly appreciate it. Prove if $x_1,...,x_n$ are natural numbers with $n\geq2$ then $x_1x_2...x_n$ is odd iff $...
1
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1answer
35 views

Show that $C_n\times K_2$ is $1$-factorable for $n\ge4$

Show that $C_n\times K_2$ is $1$-factorable (has a perfect matching) for $n\ge4.$ $\times$ means the Cartesian product. $C_n$ means a cycle where $n=$ number of vertices of the cycle. $K_2$ means the ...
0
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1answer
20 views

Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values

I'm talking specifically of population SD, where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ I have a hunch that $s$ is maximized for a certain mean $\bar{x}$ when the values in ...
3
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1answer
55 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
2
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1answer
22 views

Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
0
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1answer
17 views

Prove that the Mean Absolute Deviation cannot be greater than the greatest value in a set of positive values.

I have a feeling that this is true (only for positive values in our set, obviously) but I can't prove it or even completely convince myself that it's true. If not a formal proof (although those are ...
0
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2answers
16 views

Claim $(\mathbb{N}, \leq)$ is a discrete space, but is $(-\infty, b)$ a subbasic element?

Let $\mathbb{N}$ denote the set of natural numbers, then a subbasis on $\mathbb{N}$ is $$S = \{(-\infty, b), b \in \mathbb{N}\} \cup \{(a,\infty), a \in \mathbb{N}\}$$ Let $\leq$ be the relation on ...
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3answers
26 views

Proof by contradiction for: Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$

I am kind of stuck on a practice problem relating to proof by contradiction that goes as follows: "Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$" For the ...
1
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1answer
49 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
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3answers
24 views

Alternative perspective on mean value theorem in one-dimensional space

In my textbook, there is an alternative perspective on the mean value theorem that I don't understand. When we introduced the mean value theorem the first time, the statement looked like this: https:/...
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4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
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0answers
17 views

How to consider such a statement by contradiction ?

I am trying to do some practice problems on proofs by contradiction and am not quite sure how to interpret the statement given by: "$100$ cannot be written as the sum of 3 integers, an odd number of ...
1
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2answers
49 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...
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3answers
59 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
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0answers
14 views

trouble undestanding the proof for the therom “If x is element of N and x != 1, then there is a unique y so that x = y'.”

give the following axioms The following theorem is proven Im having trouble understanding the sentence from "if x=1 then x' element of N ..." up to "and by definition of A, x' element of A." ...
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3answers
48 views

induction clarification about the step $n+1$

Suppose i need to prove that $\frac{1}{2^2}+\frac{1}{3^2}...+\frac{1}{n^2}<1-\frac{1}{n}$ So in the step of $n+1$, the right side becomes $<1-\frac{1}{n+1}$ or is it: $<1-\frac{1}{n}-\frac{1}...
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0answers
67 views

Explanation of the proof of the expectation of a linear function.

$\newcommand{\var}{\operatorname{Var}}$ T1. LINEAR FUNCTIONS. For linear functions, the expectation of the function is the function of the expectation, and the variance of the function is the ...
2
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1answer
97 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
1
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1answer
72 views

If $G$ is a graph of order $n$ such that $\delta (G) ≥ (n-1)/2$ , then $\lambda(G) = \delta(G)$

Prove that if G is a graph of order n such that δ(G) ≥ (n-1)/2 , then λ(G) = δ(G). where δ(G)= minimum degree of the graph G λ(G)= minimum edge cuts to disconnect graph G κ(G)= minimum ...
1
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1answer
20 views

Questions about the proof: Continuous function with weak derivatives $\Rightarrow$ $C^1$

For an open set $\Omega$ of class $C^1$, suppose we have $u \in W^{1,p}(\Omega)$ and that $u$ is continuous and all the partial derivatives of $u$ are continuous. I want to show that $u$ is $C^1(\...
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1answer
24 views

Speciel orthogonal group

I'm trying to solve 3 problems on special orthogonal groups, and I need proof verification of the first 2 and help with the proof of the 3rd. Consider $SO(n)$ the set of all $n \times n$ matrices ...
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1answer
30 views

Prove for some $n$, $f^{n+1}=f^n$ and that Y is bijective.

Are these sufficient to show what is being asked? If you could confirm or provide a more efficient way to do so I would greatly appreciate it. Let $X$ be a finite set and $f:X\rightarrow X$ be a ...
1
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1answer
28 views

Can anyone explain one step in the proof of Riesz Representation theorem?

I am trying to understand the following lemma from Royden's Real Analysis which is directly used to prove Riesz Representation theorem. The book in the proof states that " when $p = 1$. We must ...
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0answers
26 views

A proof of the test of exactness for differential equations

I went through a proof of the following theorem for test of exactness of differential equations: Let the functions $M(x,y)$, $N(x,y)$, $M_y(x,y)$, and $N_x(x,y)$, be continuous on the region $R=\{(x,...
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0answers
26 views

How to prove that three different modulo 9 equations results the same sequence?

First let index sequence $ℕ_0=(0,1,2,…)$ and $n∈\mathbb{N}_{0}$. Then let: $$S_a = (-1)^n(a+bn) \text{ mod 9 } \text{ where } a = 1\text{, } b = -3$$ $$S_b = 2^n \text{ mod 9 }$$ $$S_c = F_{a+bn} \...
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1answer
54 views

Assume $r,s \in\mathbb{Q}$. Prove $\frac{r}{s},r-s \in\mathbb{Q}$ [closed]

I have attempted this proof by contradiction. Beginning with assuming to the contrary that each a and b are irrational but was not sure if I did it correctly. Any help would be greatly appreciated. ...
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2answers
52 views

(Rigor/Validity of Proof) Every sequence of reals in a compact set has a convergent subsequence

[ADDED/MODIFIED]: I began my proof with a compact set, but this was a wrong start. Although the comments are valid, I should've started with a bounded set. Because what I want to establish first is ...
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1answer
30 views

Euler's Formula: $V-E+F=2$ by using spheric triangles

I just have a question to a proof found here: https://nrich.maths.org/1384 At one point it says: As eight copies of $\triangle$ will fill the sphere without overlapping. Why this? Why can I "...
0
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1answer
12 views

An increasing function defined on a interval that is only continuous outside a countable set

Let $C$ be a countable subset of $(a,b)$. Then there is an increasing continuous function on $(a,b)$ that is continuous only on $(a,b)\setminus C$ This is an example from Royden's real analysis book....
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0answers
29 views

Confusion about the convergence of Riemann zeta function in terms of the integral

Titchmarsh wrote that $$\zeta(s)=s\int_{1}^{\infty}\frac{\left \lfloor x \right \rfloor-x+\frac{1}{2}}{x^{s+1}}\,\mathrm{d}x+\frac{1}{s-1}+\frac{1}{2}\tag{2.14}$$ using the Euler-Maclaurin summation, ...
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1answer
28 views

Show that every local homeomorphism is continuous and open therefore bijective local homeomorphism is a homeomorphism

Follow up on another question I asked recently: Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism Definition: Let $(X, \mathcal{...
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2answers
40 views

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \...
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1answer
95 views

Prove $\sum_{d\le x} \sum_{q\le x/d} \dfrac{1}{q^{\beta}} = \sum_{d\le x} \dfrac{1}{d^{\beta}} \sum_{q\le x/d} 1$?

The below texts are from the book Introduction to Analytic Number Theory by Apostol: Trying to calculate $\sum_{n\le x} \sigma_{\alpha} (n)$ for negative $\alpha$ I followed the advice of the book, ...
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2answers
46 views

Show that cuts are preserved under homeomorphism

Let $(X, \mathcal{T})$ be a topological space, assume that $X$ has no proper (not $X$ or $\varnothing$) clopen subset. Definition: A point $p \in X$ is a cut if $X \setminus\{p\}$ has a proper ...
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0answers
55 views

Quadrics intersecting the twisted cubic and a line.

I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...
3
votes
4answers
89 views

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers It is in the exercises of the AM-GM inequality chapter of a book, and that is why I believe it will be solved by ...
1
vote
1answer
12 views

Functions composition commutativity

I have to prove that $\circ$ is not, in general, a commutative operation of Funct(X,X). My approach: Let X be a set, $a,b\in X$, $a\neq b$ constants. Let $i,j \in Funct(X,X)$ with $i:X \to X,\text{ } ...
2
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0answers
46 views

Proving Schur's lemma

Schur's Lemma: Let $(\Pi_i,V_i)$, $i=1, 2$ be two irreducible representations of a group $G$, and let $\phi : V_1 \to V_2$ be an intertwiner. Then either $\phi = 0$ or $\phi$ is a vector space ...
1
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0answers
18 views

Help understanding proof of Frostman's Lemma - issue technical or termonological?

I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The ...
1
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1answer
56 views

Understanding Milnor's proof of the fact that the preimage of a regular value is a manifold

In the book "Topology from the Differential Viewpoint" (Milnor) he proves on page 11 the following lemma: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is ...