For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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2answers
30 views

Help with proof by induction?

Prove $\frac{2(n-c)}{n+1} < 2$ where c is any natural So we assume $\frac{2(n-c)}{n+1} < 2$ is true, and so far I have $\frac{2(n+1-c)}{n+2} = \frac{2n-2c+2}{n+2} = \frac{2(n-c)}{n+2} + ...
0
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0answers
26 views

geometric construction puzzle is my construction right?

I think I solved my own question of hyperbolic geometry (and circle ) construction problem but am not sure if it is correct, it looks that way but can somebody help me with prooving it? My ...
1
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0answers
40 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
3
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0answers
57 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
1
vote
2answers
52 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
3
votes
1answer
45 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
1
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1answer
21 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
0
votes
1answer
29 views

Question about eigenvalues: eigenvalue $f^2 + f = -1 \rightarrow$ eigenvalue $f^3 = 1$

I have to proof: Let $f \in End(V)$. Show that if $f^2+f$ has eigenvalue $-1$ then $f^3$ has eigenvalue $1$. My idea: If $-1$ is the eigenvalue of $f^2+f$ then there exists (per definition) a $v ...
0
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1answer
41 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
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0answers
11 views

Deducing the rules for inequalities from the order properties.

I'm trying to relearn calculus but going more in-depth this time, meaning doing all exercises and especially the ones dealing with proofs. I'm using the Thomas calculus textbook. Thomas starts by ...
2
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1answer
59 views

Learning to understand proofs faster?

There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having ...
2
votes
1answer
58 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
2
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0answers
38 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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0answers
13 views

Bijective operator has dense image

Let $T\colon C[0,1]\to C[0,1]$ be a bijective operator. My (maybe silly) question is if $T$ then has a dense image. I think I can show this in general: Let $f\colon X\to Y$ be bijective. Consider any ...
0
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2answers
45 views

Using epsilon -delta definition to show that $\lim_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $

I'm trying to use the $\varepsilon$-$\delta$ definition to show that $\lim\limits_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $. I already know how to prove continuity for cubic root using ...
17
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3answers
664 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
1
vote
0answers
19 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...
0
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0answers
28 views

A question about open “balls”

I've been recently learning Topology and I'm struggling to visualize open balls. For instance, on $\mathbb{R}^2$ and $\mathbb{R}^3$ given a metric like say $d_\infty(x,y)=\sup\{|x_1-y_1|,|x_1-y_2|\}$ ...
2
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0answers
33 views

Please check these proofs for sets

I would appreciate the insight again for a couple of proofs since I'm learning. These are homework problems in so much as they are problems from the textbook. They are not required by my professor. ...
1
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0answers
33 views

Convergence of geometric shapes

Let $c$ be a closed geometric shape. Let $P$ and $Q$ be two points in $c$. Let $S$ be the family of all closed squares. Let $s_1, s_2,...$ be a sequence of shapes from family $S$ such that for every ...
1
vote
1answer
28 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
1
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0answers
63 views

Tim Chow's proof that the moser-number is much smaller than grahams number

The link here shows a proof from Tim Chow that the moser-number is much smaller than grahams number. I do not understand the inequality 3^^...^^3 (3^^^^^3×2-1 arrows) << G 2 What does G 2 ...
0
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1answer
28 views

Projection operator image and kernell

Proof that: Linear operator ($P:X\to X$) is projective ($P^2=P$) IFF $\exists$ direct sum decomposition of X ($X=V\oplus U$), such that $\forall u\in U:Pu=u$ and $\forall v\in V:Pv=0$. My proof: ...
0
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2answers
43 views

Simple inequality, proof

There is a following inequality ...
2
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4answers
63 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
1
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1answer
44 views

Ascending chain condition and ring homomorphism

Let $f : R \to S$ be a surjective ring homomorphism between two integral domains. Could anyone advise me on how to prove/disprove the following statements: If $R$ satisfies the ascending chain ...
3
votes
1answer
48 views

Functional form of a solution to a Differential Equation (Euler-Lagrange)

Let $f=f(q(t),\dot q(t),t)$, where $q(t)=\{q_1(t),...,q_N(t) \}=\{q_{a}\}_{a=1}^N$ and $\dot q:=\frac{dq}{dt}$. I want to show that if the following equations (Euler-Lagrange) are satisfied ...
0
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0answers
31 views

The unique model of cardinal $\kappa$ of a $\kappa$-categorical countable theory is saturated.

Let $T$ be a $\kappa$-categorical ($\kappa \geq \aleph_1$) first-order theory in a countable language $\mathcal L$. I try to prove that its unique (up to isomorphism) model of cardinal $\kappa$ is ...
2
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3answers
53 views

Determine if the set is a basis for the vector space

There is a linearly independent set of vectors in the vector space $V$, given by $\{v_1, v_2,...v_k\}$. We have to show that the set $\{v_1, v_2,...v_{k-1}\}$ cannot be a basis for V. It is clear ...
1
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1answer
50 views

Prove thoroughly: If the degree of all vertices is greater or equal to $\frac{|V| - 1}{2}$, then the simple graph is connected.

I am struggling to write a good, thorough proof. The proof is supposed to be logically rigorous, correct and complete (e.g. no hidden assumption). Moreover, style is important - the proof should be ...
0
votes
1answer
23 views

Question about proof of Krein-Milman Theorem.

I am in the middle of working through the details of the proof of the Krein-Milman theorem in Rudin's Functional Analysis (Theorem 3.23), and I am stuck on one detail. I will state the theorem and ...
1
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1answer
30 views

Proof that 6 divides $a \in \mathbb{Z}, a(a^2 - 7)$

I am trying to prove a question from my tutorial sheet, is this an acceptable proof? Six cases exist: $$a,k \in \mathbb{Z}, a(a^2 - 7) = 6k \\\text{Proof:}\\ a = 0 \mod 6 \longrightarrow a^2 = 0 \mod ...
3
votes
3answers
87 views

Please check this proof

I'm taking Discrete Math this semester. While I understand the mechanics of proofs, I find that I must refine my understanding of how to work them. To that end, I'm working through some extra ...
0
votes
1answer
34 views

Induction to prove that something is not true?

This is maybe a very basic question, but I have never seen it done before. Can you use induction to prove that something is not true? In particular if something does not hold in dimension n=1, can I ...
0
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0answers
33 views

Proving this property of a tournament graph by induction?

I am working on practicing proofs by induction. Can you please take a look at the proof below and tell me if I proved it correctly? I am particularly worried about the inductive step. Definition ...
2
votes
2answers
34 views

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$ My Proof: Ten cases exist, yielding the following equalities: $$(1\mod{10})^2 = 1\mod{10}$$ $$(2\mod{10})^2 = 4\mod{10}$$ ...
0
votes
1answer
22 views

Normal Subgroups proof help

Show that if $H$ is a subgroup of $Z(G)$, then $H$ is a normal subgroup of $G$. This is what I have so far. proof: Suppose $H$ is a subgroup of $Z(G)$. Let $h$ be in $H$ and $g$ in $G$. Then ...
1
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3answers
46 views

Error for graph Theory proof

I am looking for an error in the proof but I am not certain about it. Pretty sure it has something to do with how there is not always a cycle of length 3. Theorem 1. For every (undirected) graph ...
0
votes
0answers
28 views

Verifying $G*H$ Has Trivial Center and Elements of Infinite Order

Hypothesis: Let $G \ne H$ denote two non-trivial groups. Goal: Show that $G * H$ has a trivial center (hence is non-abelian) and contains an element of infinite order. Is my attempted proof below ...
2
votes
1answer
46 views

Proving that $f$ is integrable if limt exists

First of all please keep in mind that i want a hint not a full solution. Let $P$ be partition on $[a,b]$. Let $h(P)=\max_{i\leq j \leq n}|x_j - x_{j-1}|$. Let $c$ = $(c_1,c_2,\dots,c_n)$ such that ...
1
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0answers
25 views

Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)

Hypothesis: Let $G$, $H$, $G'$, and $H'$ be cyclic groups of orders $m$, $n$, $m'$, and $n'$ respectively. Goal: Show that if $G * H$ is isomorphic to $G' * H'$ then $m = m'$ and $n=n'$ or else $m ...
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0answers
35 views

Belief Operator

Appealing to the Kripke $S5$ system, belief operator is defined in the following way, $B_iA=\left\{\omega \in Y: \pi_i(A\mid\omega)=1 \right\}$ where $\pi_i:Y\to\Delta(Y)$. The meaning of this given ...
0
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0answers
26 views

Help to prove equation involving the Dirac Delta

Consider the functions $w(E)$, $W(E)$ and $\Omega(E)$ $\begin{aligned} w(E) &= e^{-\beta E} \\ W(E) &= \int \delta(E - H(\mathbf{x}))w(H(\mathbf{x}))\mathrm{d}\mathbf{x} \\ \Omega(E) &= ...
2
votes
1answer
96 views

Is the following set stratified (and why not) in New Foundations?

notation: $Id=\{\langle x,y\rangle : x=y\}$ (identity relation) $X[y]$ (image of an element y under a relation X) the set I am asking for is: $Z=\{\langle x,y\rangle : \neg \exists k\; y \in k ...
0
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0answers
52 views

Ascending chain condition holds in a lattice implies every ideal is principal

Proof: Suppose for contradiction that ACC holds for a lattice L, but there exists an ideal which is not principal. Thus $\exists$ I $\subset$ L s.t. ~$\exists$x: x $\le$a for a$\in$L. Thus $\exists$I ...
0
votes
1answer
36 views

Proof of the 'second' triangle inequality

I am trying to prove the 'second' triangle inequality: $$||x|-|y|| \leq |x-y|$$ My attempt: $$----------------$$ Proof: $|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = ...
3
votes
2answers
53 views

Help finishing proof via induction for a summation

So I have to prove the following equation using induction for n >= 2: $$ \sum\limits_{i=1}^n 4/5^i < 1 $$ However the question asks me to prove something stronger such as this: $$ ...
0
votes
1answer
39 views

set theory, infinite set proof, is it alright?

$\Bbb{N}$ is the natural numbers set (included $0$). let be $n\in\Bbb{N}$, $A_n = \{x\in \Bbb{N}|0\leq x \leq n\}$ prove of disprove: $$\forall n,k \in \Bbb{N},\exists m \in\Bbb{N}(|A_m - A_n|=k)$$ ...
4
votes
1answer
63 views

Localization of Coordinate Rings: $\mathbb C[V_f] = \mathbb C[V]_f$.

Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb ...
1
vote
2answers
54 views

l'Hopitals rule - is my working correct?

Is anyone able to help me with this question on l'Hopital's rule? Use l'Hopital's rule to find the limit of the sequence $\{a_n\}_{n=1}^\infty$ with $n$-th term $\displaystyle a_n = ...