Questions tagged [proof-explanation]
This tag is for readers who ask for explanation and clarification of some steps of a particular proof.
0
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1answer
18 views
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 ...
1
vote
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 ...
0
votes
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
votes
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 $...
1
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0answers
24 views
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
$$0 \to H_n(X^n) \stackrel{j_n}\to H_n(X^n, X^{n-1}) \stackrel{\partial_n}\to H_{n-1}(X^{n-1})\to?$$
...
0
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2answers
28 views
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 ...
3
votes
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.
...
1
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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 $\...
1
vote
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/...
1
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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 $$\bigcup_{n=1}^{+\infty}\bigg(a,b-\frac{1}{n}\bigg]\subseteq (a,b),$$
...
0
votes
1answer
32 views
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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votes
1answer
13 views
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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2answers
23 views
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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votes
2answers
28 views
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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0answers
24 views
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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0answers
43 views
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| = |...
0
votes
0answers
14 views
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 ...
1
vote
2answers
52 views
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 ...
1
vote
1answer
16 views
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 ...
0
votes
2answers
26 views
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.
5
votes
4answers
83 views
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 ...
0
votes
1answer
15 views
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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votes
0answers
17 views
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.
-2
votes
0answers
20 views
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 ...
0
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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=...
2
votes
1answer
31 views
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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0answers
44 views
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
30 views
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,
...
0
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1answer
30 views
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 ...
1
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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 ...
7
votes
1answer
44 views
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$.
...
0
votes
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(...
0
votes
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 ...
1
vote
1answer
24 views
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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0answers
16 views
The question about proof of the uniqueness lemma
I can't understand why marked with green is right. Thanks in advance for any hints.
3
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1answer
38 views
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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votes
2answers
39 views
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 ...
0
votes
1answer
23 views
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 ...
0
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1answer
16 views
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 ...
0
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0answers
27 views
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:
0
votes
2answers
41 views
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?
0
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1answer
47 views
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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votes
2answers
24 views
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 $n! > n^2$?
1
vote
1answer
28 views
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 ...
1
vote
3answers
32 views
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 ...
1
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2answers
32 views
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 ...
1
vote
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....
3
votes
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 ...
0
votes
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_{\...
