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Questions tagged [proof-explanation]

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

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Showing a measure finitely additive but not countably additive

While reading Vector Measure from Diestel's book I find that Considering any Hahn-Banach extension $T$ to $L_{\infty}[0,1]$ of point mass functional on $C[0,1]$ we can construct a measure $F$ defined ...
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1answer
14 views

The limit of a strongly convergent sequence of linear bounded operators from a Banach space to a normed space

Let $X$ be a Banach space and $Y$ be a normed space. If the sequence $\{T_n\}$ of bounded linear operators from $X$ into $Y$ is strongly convergent. Then there exists a bounded linear bounded operator ...
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1answer
27 views

Unclear on step #2 of the MathWorld definition of the Reimann Prime Counting Function

I was reading through the MathWorld article on the Reimann Prime Counting Function. The first step in the definition is clear to me: $$f(x) = \sum_{p^v < x \text{ and p prime}} \frac{1}{v}$$ ...
2
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1answer
29 views

Circularity of LEM, principle of explosion, and $\lnot \lnot$ elimination

Consider the following proof of the principle of explosion using $\lnot \lnot$ elim: |Assume $p \land \lnot p$ $\quad$|$p$ (from $\land$ elim) $\quad$|$\lnot p$ (from $\land$ elim) $\quad$|Assume $...
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0answers
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Successive maps in exact sequence leads to $0$. Celluar Homology

This comes from Hatcher's Algebraic Topology book on page 139 He says that the map in the diagram $$0 \to H_n(X^n) \stackrel{j_n}\to H_n(X^n, X^{n-1}) \stackrel{\partial_n}\to H_{n-1}(X^{n-1})\to?$$ ...
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2answers
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If $A:X \rightarrow X$ is a linear bounded operator then $e^{A}:X \rightarrow X$ is a linear bounded operator

Let $L(X)$ be the space of all linear bounded operators on $X$ under the operator norm. What I got so far was, since we know that $L(X)$ is a Banach space thus every absolutely convergent series is ...
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4answers
46 views

All positive integers a and b such that $a! + 6 = b^2$

I'm not sure how to approach proving solutions for this problem. I wrote a python program which shows a must be $\geq 30$, but I don't understand why. ...
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1answer
17 views

A family of sets that is not a $\sigma$-algebra

Let $X$ be a not empty set. I must prove that the family $\mathcal{A}$ defined as follows $$\mathcal{A}=\bigg\{A\in\mathcal{P}(X)\;|\;A\;\text{is finite or $A^c$ is finite}\bigg\}$$ it is not a $\...
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1answer
42 views

Show that $G_i/G_{i+1} \twoheadrightarrow (G_i +N)/ (G_{i+1} +N)$.

From Aluffi's book of Algebra: in which it refers to Theorem 7.12 : [Red underlined:] I can't see any connection with Theorem 7.12! Actually I don't know how the $+$ appears. How to show that $G_i/...
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1answer
20 views

Prove that $(a,b)=\bigcup_{n=1}^{+\infty} \bigg (a,b-\frac{1}{n}\bigg]$

Let $a,b\in\mathbb{R}$ with $a<b$ we must prove that $$(a,b)=\bigcup_{n=1}^{+\infty} \bigg (a,b-\frac{1}{n}\bigg].$$ Obviously $$\bigcup_{n=1}^{+\infty}\bigg(a,b-\frac{1}{n}\bigg]\subseteq (a,b),$$ ...
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1answer
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Tricky question about limit of a function

In my Calculus home work assignment I get the following tricky question about the limit of following function: Define a function $f : \mathbb{R} \rightarrow \mathbb{R}$ as follows: if $x \in \mathbb{...
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1answer
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Determining properties of row spaces of two column-equivalent matrices

Given matrix $$A = \begin{pmatrix} 1 & r & 1 & 1 \\ 1 & s & 1 & 2 \\ 1 & t & 1 & 3 \\ 1&0&0&1\end{pmatrix}$$ and let $B$ be the matrix obtained from $A$ ...
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Show that if $z$ is a complex number such that $z^5$ + 5$z^2$ + 7 = 0, then |$z$| > 1.

How do I show that if $z$ is a complex number such that $z^5$ + 5$z^2$ + 7 = 0, then |$z$| > 1. I thought maybe try by contradiction i.e letting |$z$| $\leq$ 1. But then I ended up getting |$z^5$ + ...
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2answers
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Show that if $z^n$ + $z^{n-1}$ + … + $z$ + 1 = 0 then $z$ $\neq$ 1 and $z^{n+1}$ =1.

Show that if $z^n$ + $z^{n-1}$ + ... + $z$ + 1 = 0 then $z$ $\neq$ 1 and $z^{n+1}$ =1. I thought I had managed to show that $z$ $\neq$ 1 since if $z$ = 1 then $z^n$ + $z^{n-1}$ + ... + $z$ + 1 = 1 ...
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A question of the proof of Th 8.1.4 in Real Analysis and Probability by Ash

In Real Analysis and Probability, Th 8.1.3 used a proposition that Let $g$ be a continous map from $\mathbb{R}$ to $\mathbb{C}$, with compact support, that is for some $T>0$, $g=0$ off $[-T,T]$....
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Mathematical proof

i have problem with this mathematical proof. \begin{align*} 5 \leq 4|x-1| + |2-3x| \end{align*} I have these 3 properties, in the field of real numbers \begin{gather} \forall x \forall y\, |xy| = |...
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A function analytic in the unit disk belongs to the class Nevanlinna if and only if it is the quotient of two bounded analytic functions

I'm trying to understand a part of this proof from Duren, in the converse, I don't see it clear when it says "by analytic completion of the Poisson Formula,..." and then the result; I tried to prove ...
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2answers
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solving natural log inequality

How can I show that $0 \le (\sum_{x=1}^{n}\frac{1}{x})-ln(n) \le 1 - \frac{1}{n}$ Do I raise both sides by $e$ or perhaps take integral of both sides? If so, I'm still not quite sure where to go ...
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1answer
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Prove the comparability theorem for well ordered sets using transfinite induction

My question is about the same proof discussed here, but I am confused about more than this question addresses. In Naive Set Theory, Halmos phrases the comparability theorem as follows: The ...
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2answers
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solving integral inequality

How can I show that $0 \le \int_n^{n+1}\frac{1}{n}-\frac{1}{x}dx\le \frac{1}{n}-\frac{1}{n+1}$ I think I need to take the log to solve this, but I'm not quite sure.
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4answers
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Does $a_{n}/a_{n-1}$ converge to the golden ratio for all Fibonacci-like sequences?

Yesterday a friend challenged me to prove that $$\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}=\varphi\; ,$$ where $\varphi$ is the golden ratio, for the Fibonacci series. I started rewriting the ...
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1answer
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Equivalent forms for a product notation

Context: See "2 Hoeffding’s Inequality" in : http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf My particular question arises within 'section 2 Hoeffding's Inequality' is: $$ e^{-tn\varepsilon }\...
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proving the converse of Eulers totient multiplicative function [on hold]

I am trying to prove that if φ(mn)=φ(m)φ(n) then m,n are relatively prime.
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Prove that $E$ is not well-defined [on hold]

$E$ defined by $[x] \oplus [y] = [x + y]$. Prove that $E$ is not well-defined. I do not really understand how this function could not be well-defined. Any chance someone could show me an example of ...
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0answers
13 views

Polynomial Estimation Lemma

I'm struggling to understand the concept behind polynomial estimation. I have applied an example to this method but still unsure on how or why the conclusion is reached. I'm applying the steps to ...
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1answer
12 views

Borel-Cantelli Theorem of a Finite Series of Independent Events

Let $\{A_n\}_{n=1}^{\infty}$ be a independent sequence of events such that $\sum_{n=1}^{\infty}P(A_n) <\infty$, then $P(A_n i.o.) =0 $. We have that $P(A_n i.o.) =\bigcap_{m=1}^{\infty} \bigcup_{n=...
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1answer
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Question on proof of every finitely generated vector space has a basis

The following is a proof of the theorem given in Curtis's Linear Algebra book: First we consider the case in which $V$ consists of the zero vector alone. Then the zero vector spans $V$ but cannot ...
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Proving by Contrapositive (specific question within)

I'm having issues coming up with a contrapositive proof for the following question. As far as I know, a proof by contraposition is based on the following : $\overline Q \to \overline P \equiv P \...
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2answers
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Rudin functional analysis, theorem 2.11 (Open mapping theorem)

Going through the proof of such theorem, there are few bits I don't understand. I'll write down the proof and in in the middle I'll add some comments. Statement: Suppose (a) X is an F-space, ...
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1answer
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When can I assume that an open interval is an open set in topology?

I am confused as to what axioms are at my disposal. I am using Viro, Yvanov, Netsvelaev Elementary Topology Problem Book and it uses an IBL-Approach. The definition I have so far are "elements of a ...
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1answer
18 views

Signature of matrix associated with q

For $\alpha\in\mathbb{R}$, let $q(x_1, x_2) = x_1^2 + 2\alpha x_1x_2 + \dfrac{1}{2}2x_2^2$, for $(x_1, x_2) \in \mathbb{R^2}$.Find all values of $\alpha$ for which the signature of $q$ is 1. The ...
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1answer
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How many ultrafilters there are in an infinite space?

I'm stuck with the next exercise from the book Rings of Continuous Functions by Gillman. If $X$ is infinite, there exist $2^{2^{|X|}}$ ultrafilters on $X$ all of whose members are of cardinal $X$. ...
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1answer
35 views

Reduction formula for $\int\frac{dx}{(ax^2+b)^n}$

I recently stumbled upon the following reduction formula on the internet which I am so far unable to prove. $$I_n=\int\frac{\mathrm{d}x}{(ax^2+b)^n}\\I_n=\frac{x}{2b(n-1)(ax^2+b)^{n-1}}+\frac{2n-3}{2b(...
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1answer
27 views

Proof of Froda's Theorem (explanation)

Theorem: Let $f$ be a real valued function of real variable defined on open interval $(a,b)$ and let $f$ be monotonic. Then the set of all discontinuities is at most countable. I would like an ...
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1answer
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Completely lost on using strong induction for this proof regarding a recursive algorithm.

T(n) = 1 when n $\le$ 10 T(n) = T($\lfloor\frac{n}{5}\rfloor$) + 7 T($\lfloor\frac{n}{10}\rfloor$) + n when n $\gt$ 10 Prove by strong induction that, for all n ≥ 1, we have T (n) ≤ 10n. ...
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The question about proof of the uniqueness lemma

I can't understand why marked with green is right. Thanks in advance for any hints.
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1answer
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In this visual proof for the law of cosines, why are the products of subsegments of two intersecting chords equal?

The first line of the visual proof below states that $$(2a\cos\theta-b)b=(a-c)(c+a)$$ I understand the line segments represented by each part of the equation, but what makes the equation true? In ...
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2answers
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A course of Pure mathematics incorrect

I thought I'd go through 'A Course Of Pure Mathematics' by G H Hardy And I came to this part: If the reader will mark off on the line all the points corresponding to the rational numbers whose ...
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1answer
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Is it possible to simulate a seven sided die without rerolling?

In front of me, I have an arbitrary number of four-sided, six-sided, eight-sided, ten-sided, twelve-sided, and twenty-sided dice. Using any number and combination of these, is it possible to exactly ...
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1answer
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How can I prove that two null spaces are equal by using the existence of an invertible operator?

Suppose $W$ is finite dimensional and $T_1, T_2 \in L(V,W).$ Prove that null($T_1$) $=$ null($T_2$) if and only if there exists and invertable operator $S\in L(W)$ such that $T_1 = S\circ T_2$. I'm ...
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Exercise 4.6.11 (in Petrovic)

The question is given below: Could anyone give me a hint for its solution? or can I ask how is the proof different from the proof of the following theorem:
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2answers
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A difficulty in understanding a proof for L'Hospital's rule (in Petrovic)

The theorem and its proof is given below: But I could not understand why $F$ & $G$ are defined as thought, could anyone explain this for me please?
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1answer
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Boolean algebra - prove $x_1 = x_2$

I'm trying to prove that two boolean algebra expressions are equivalent: $x_1 = a'b'c + bc' + ac + ab'c$ $x_2 = b'c + bc’ + ab$ I got up to here: LHS $a'b'c + bc' + ac + ab'c$ RHS $= (a + a')c +...
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How do I prove the statement below by the method of mathematical induction? [duplicate]

If $n ∈ N$ and $n \ge 4$ , how do I prove by the method of induction that $n! > n^2$?
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1answer
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Durrett Probability Exercise 2.3.9

Durrett 2.3.9: Let $d$ be a metric on the space of random variables on a given probability space. Show that the metric $d$ is complete. That is if $d(X_m, X_n)\to 0$ as $m,n\to\infty$, then there is a ...
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3answers
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Some questions about proving with induction

I have a good understanding of how to use induction, but these two proof are really puzzling me. 1)Use mathematical induction to prove that for every positive integer $n$ greater than $23$, there ...
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2answers
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If the set of all upper bounds of $A$ and $B$ are equal and $\sup A$ exists, then $\sup B$ exists and $\sup A=\sup B$.

Caution: Axiom of Completeness is not assumed here. Before reading my attempt, I want you to think up a proof of your own. Here is my attempt: This concludes my proof. Two questions: How do I show ...
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4answers
40 views

Prove that the sequence $a_n = \sum_{k=0}^n \frac{1}{k!}$ is convergent. [duplicate]

Prove that the sequence $a_n = \sum_{k=0}^n \frac{1}{k!}$ is convergent. I am using the theorem: All bounded monotone sequences converge. So i need to prove it is bounded and monotone. $a_1=1,a_2=1....
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2answers
53 views

Understanding the proof of $V=L \rightarrow \Diamond$

I am trying to understand the proof that $\Diamond$ holds in the constructible universe. I am following Kunen's Set Theory, where the proof is on pages 230-231 (the more recent, 2011 edition). I am ...
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2answers
27 views

What are $I_1,I_2,…I_k$? How it is related to $I$?

Let $\{(X_\alpha,\mathscr T_{\alpha}):\alpha \in \Lambda\}$ be an indexed family of Hausdorff spaces such that each $X_\alpha$ has atleast two points, and let $X=\Pi_{\alpha \in \Lambda} X_{\...