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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain your proposed proof/solution.)

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Implication of stability of Van der Pol oscillator.

Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu=0$. We convert it into a dynamical system, $$\dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \equiv (...
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2answers
13 views

Proof by contradiction - am I correct?

I am trying to learn some discrete mathematics alongside my arts course to try and expand my knowledge. I have this question: Prove that if $x$ is irrational then $\frac{x+1}{x-1}$ is irrational. My ...
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1answer
21 views

Prove $\lim\limits_{x\to\frac52}\frac1{4x-8}=0.5$ via delta epsilon

prove $\lim\limits_{x\to\frac{5}{2}}\frac{1}{4x-8}=0.5$ via delta epsilon I have the following $|\frac{1}{4x-8}-\frac{1}{2}|<\epsilon$ $|\frac{5-2x}{4x-8}|<\epsilon$ which I got after ...
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0answers
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$f(x)=\frac{10x}{x^2+1}$ Find greatest value of $\delta$ accurate to three decimal places that $0<|x-1|<\delta$ guaruntee $|f(x)-5|<0.01$

$f(x)=\frac{10x}{x^2+1}$ Find greatest value of $\delta$ accurate to three decimal places that $0<|x-1|<\delta$ guaruntee $|f(x)-5|<0.01$ My question- How do I even start?!
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1answer
13 views

Almost sure convergence and bounded supremum

Suppose $\lim_{n→\infty} X_n=X$ a.s. and $|X|<\infty$ a.s. Let $Y=\sup_n|X_n|$. Show that $Y<\infty$ a.s. I stumbled across this problem while reading through Jacod and Protter's Probability ...
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41 views

Compute $\lim \limits_{n\to \infty} n^{1/n}$

Two part question that I want to make sure I did correctly. a) Let $x_n = \sqrt[n]{n} - 1$. Use the fact that $(1 + x_n)^n = n$ to show that $x_n^2 \leq \frac{2}{n}$. Hint given to use the binomial ...
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0answers
14 views

If $R$ is a ring such that $x^2=x$ $\forall x \in R$ and $I$ is a prime ideal. Show that $R/I$ has two elements

If $R$ is a ring such that $x^2=x$ $\forall x \in R$ and $I$ is a prime ideal. Show that $R/I$ has two elements. $R/I = \{ r+I:r\in R \}$ Let $a \in R$ if $a\in I$ then $a+I = 0+I=I$. if $a\notin ...
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3answers
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Proving $x^2 - 4y^2 = 7$ has no natural numbers

Ok so I needed to prove this by contradiction. Let $P:~x^2 - 4y^2 = 7$ and $Q:~x,y$ are not natural numbers Note that $N$ does not include $0$ OK to begin to prove by contradiction we are given $P$...
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Proof of Generalised ratio test

Is the following argument correct? Proposition. Suppose $\{x_n\}$ is a sequence and suppose for some $x\in\mathbf{R}$, the limit $$L:=\lim_{n\to\infty}\frac{|x_{n+1}-x|}{|x_n-x|}$$ exists and $L<1$...
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0answers
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Checking out my demonstration on metric spaces

Let $(X,d)$ a metric space. Prove that $(X,d)$ is complete if, and only if, for all sequence of closed embedded non-empty sets $(F_{n})$ in $X$ such that $\lim_{n\rightarrow \infty}\textrm{diam}(F_{n})...
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1answer
20 views

Proof verification: the homology groups of the cube.

I was so proud of this solution about finding the homology groups of the cube $I\times I \times I$ but then a friend of mine make me feel uncomfortable about it. My "proof" is this: The homology ...
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0answers
24 views

Proof by contradiction for all non zero integers a and b

Proposition For all nonzero integers $ a $ and $ b $, if $ a + 2b \ne 3 $ and $ 9a + 2b \ne 1 $, then the equation $ ax^3 + 2bx = 3 $ does not have a solution that is a natural number. Proof We ...
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Show that a function cannot be expressed as a strictly increasing transformation of another function

Consider a function $\Phi: \mathcal{I}\subseteq \mathbb{R}\rightarrow \mathbb{R}$. Suppose (a) $\Phi(0)=0$ (b) $\Phi(1)=1$ with $0\in \mathcal{I}$ and $1\in \mathcal{I}$. Consider a function $\...
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0answers
31 views

Finding density and distribution of $X=\tan(\theta)$?

Here is a problem I'm trying to solve and wanted to check if my solution was ok. A searchlight is distance 1 from a wall. Let $Q$ denote the point on the wall directly opposite it and assume it scans ...
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0answers
13 views

Numbers $n$ such that $\sigma_1[n(n+1)/2]=n^2+n-2.$

Question: If $n$ is a positive number greater than one, I write $a_n=n(n+1)/2$ and denote the prime Omega counting function$^1$ on $a_n$ by $\Omega\left(a_n\right).$ Let $b_n={\gcd\left[a_n, 2^{\Omega(...
2
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1answer
23 views

A Lemma in Transfinite Recursion Theorem

Transfinite Recursion Theorem: Let $G:V\to V$ be a class function. Then there is a unique function $F:\operatorname{Ord}\to V$ such that $$\forall \alpha\in \operatorname{Ord}:F(\alpha)=G(F\...
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2answers
28 views

Proving a contrapositive of the ratio test for sequences.

Let $(a_n)$ be a sequence s.t. $a_n\rightarrow L\in\mathbb{R}\setminus{\{0\}}$, let $\forall n\in\mathbb{N},a_n>0$. We wish to show that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1$. (From definition). ...
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2answers
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Calculating Null T and Range T for the linear transformation $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$?

I have a doubt regarding the calculation of range of a linear transformation. I will explain my doubt with an example. Suppose, $T:R^3 \to R^3 \ni$ $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$ is a Linear ...
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1answer
56 views

Open set in $\Bbb{R}^4$ [ Direct Proof]

Prove: The set $$U:=\Big\{(a,b,c,d) \in \Bbb{R}^4: \vert ad-bc \vert >1\Big\}$$ is open in $\Bbb{R}^4$ . This question is already asked today [see this post]. The answer in this post involves "...
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1answer
9 views

Dual Transformation is 1-1 implies tranformation is onto

I Need to show that $T:V_1\to V_2$ is onto iff $T^*:V_2^*\to V_1^*$ is one one. Here $V_1,V_2$ are vector spaces and $T$ is linear transformation I do not want to use rank/transpose argument. Left to ...
2
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3answers
35 views

Proof of $f^{-1}(A) \cap f^{-1}(B)= f^{-1}(A \cap B)$

First let me say I am aware of the other threads on this result. The reason for me making this thread is to find out whether or not my proof/proof attempt is correct. The problem stated in full ...
2
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1answer
22 views

Show that $\log_b{(y\cdot z)} = \log_b(y)+log_b(z)$

For $a,b > 1$ and $y,z > 0$, show by using only the power rules and the definition $x = \log_b(y) \Leftrightarrow_{df} y = b^x$ that $\log_b{(y\cdot z)} = \log_b(y)+log_b(z)$. I don't get how ...
5
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1answer
29 views

Homotopic maps induce the same homomorphism for reduced homology groups

This is an exercise from Hatcher's Algebraic Topology book: http://pi.math.cornell.edu/~hatcher/AT/AT.pdf Exercise 2.1.13, pg. 132: Verify that $f≃g$ implies $f_∗=g_∗$ for induced homomorphisms of ...
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1answer
12 views

Sequential Compactness implies compactness.

Proofs I saw used the Lebesgue number or Lindelof property. I used total-boundedness and I wonder if this is correct. Proof attempt: By a known result. If $X$ is sequentially compact then it is ...
5
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1answer
52 views

Why isn't Legendre's Conjecture resolved by the work done by Nair and Hanson in relation to Least Common Multiples?

I recently discovered the work done by Hanson and Nair. As I worked through Hanson's proof, an argument occurred to me regarding Legendre's Conjecture. Clearly, my argument is wrong. It is too ...
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0answers
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Continuity and Uniform Continuity on Dense Subset Implies Uniform Continuity

Problem: Let $(X,d_{1})$ and $(Y,d_{2})$ be metric spaces. Suppose that $A$ is a dense subset of X, $X\xrightarrow{f} Y$ is continuous, and $f$ is uniformly continuous when restricted to $A$. Prove ...
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1answer
24 views

If $f$ is continuous then $G_f \subset M \times \mathbb{R}$ is closed.

Consider a function $f: M \rightarrow \mathbb{R}.$ The graph of $f$ is the set $$G_f := \{(x,y) \in M \times \mathbb{R} : y = fx\}.$$ Prove that if $f$ is continuous then $G_f \subset M \times \...
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2answers
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Proving $A\subset B\implies |A|\leq |B|$

I would like to prove the following: Let $A,B\subset\mathbb{R}$ be non-empty finite sets. Prove that if $A\subset B$, then $|A|\leq |B|$. We are also given the following theorem: Theorem $1....
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1answer
48 views

How many automorphisms does $D_5$ have?

Here is my solution: Recall how $D_5$ is defined. We have 4 non-identity rotations: $$ T, T^2, T^3, T^4, $$ with $T^0 = 1$. We have the reflection $S$, such that $S^2 = 1$. To connect reflected ...
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0answers
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How to define sequence inf{an} and sup{an}

How do we define the subsequences of supremums an infinimums of a given sequence? For example, here is one of the problem I was working on and I used the theorem of that if a sequence has a limit L, ...
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2answers
73 views

Is $\int_0^1 f(x)dx$ bounded if $\int_0^1 f(x)dx - \int_0^1 f(y)dy$ is bounded?

Can we conclude that $\int_0^1 f(x)dx$ is bounded if $\int_0^1 f(x)dx - \int_0^1 f(y)dy$ is bounded? I have a question that asks if $f$ is measurable on $(0,1)$, and $f(x)-f(y)$ is integrable over ...
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2answers
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How to solve $2y′′+8y′+80y=F(t), y(0)=0, y′(0)=0$?

I am trying to solve the initial value problem, $2y′′+8y′+80y=F(t), y(0)=0, y′(0)=0$ where $F(t)=20e^{-t}$ But I am unable to do it. I got the answer of $2Ae^{-t}=20e^{-t}$ But this is wrong I ...
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2answers
18 views

Translate the following English sentences into symbolic sentences with quantifiers.

this is my solution of my homework, is that true 100%? Please if there is any mistake tell me because my professor is so careful. Thanks. (Sorry, I don’t speak English well) Translate the following ...
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1answer
42 views

Proving the pullback of monics is monic.

I mostly just want to double-check my reasoning in my proof. For clarity's sake, the diagram we are working with is where $m$ is monic, the top-left corner is a pullback, and we wish to show that $m'$...
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0answers
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Proving $\cup A_m$ $\subseteq$ B

Trying to prove $\cup A_m$ $\subseteq$ B where $$\cup A_m=\{(x,y) \in R^2|0 \lt y \le \frac{-m}{4}x^2+m\}$$ $$B=\{(x,y) \in R^2 | -2 \lt x \lt2 \land y\gt0\}$$So my first step in proving $\cup A_m$ $\...
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What configuration will let you hold most popcorn with a piece of paper? [on hold]

What configuration will let you hold the most popcorn with a piece of paper
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18 views

Constructing a Borel Measure

I'm working on the following problem: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Borel measurable. Let $\nu(E) = \int_Ef\,d\mu$. Show that $\nu$ is a Borel measure. Showing this is a measure ...
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1answer
24 views

Smooth extension of a smooth map on an non-empty open subset of a manifold to the whole manifold.

Proposition: Suppose $M$ is a smooth manifold and $\emptyset\neq U\subset M$ is open and $f:U\rightarrow \mathbb{R}$ is a smooth function. Then $f$ does not necessarily extend smoothly to M. Proof: (...
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1answer
25 views

The non-empty intersection of two open discs contains an open disc.

Is the following argument correct? Let $D_1$ and $D_2$ be any open discs in $\mathbf{R}^2$ with $D_1\cap D_2\neq\varnothing$. If $(a,b)$ is any point in $D_1\cap D_2$, show that ther exists an open ...
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1answer
46 views

Is this proof correct or am I trying to prove a falsehood?

Suppose that we have three possible outcomes $\alpha$, $\beta$ and $\gamma$ (if that helps, suppose they are monetary amounts) and some threshold $x$. Then, we also know that \begin{gather*} x\...
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3answers
43 views

Validate the proof that the sequence $x_n = \sum_{k=1}^{n} {1\over n+k}$ is bounded.

Let $n \in \mathbb N$ and: $$ x_n = \sum_{k=1}^{n} {1\over n+k} $$ Prove that $x_n$ is a bounded sequence. I'm wondering whether the proof below is valid. Since $n \in \mathbb N$ we have that $...
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0answers
33 views

A proof of Transfinite Induction

This is my proof of Transfinite Induction by filling the gaps in my textbook. It would be great if someone help me verify if i correctly understand what is meant by the authors! Let $P(\alpha)$ is ...
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1answer
26 views

Proving that a unit disc is open in $\mathbf{R}^2$.

I am trying to prove part $(i)$ here. The following is my attempt at the problem is it correct? Proof. Let $\alpha = (a,b)\in D$ and $\beta = (x,y)\in R_{(a,b)}$ and $O = (0,0)$, then given the ...
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1answer
16 views

Show that $L(O)=O$ if $L$ is a linear map from one vector space to another

From S.L linear algebra: Let $L:V \rightarrow W$ be a linear map from one vector space to another. Then show that $L(O)=O$. ($O$ is a null vector). There are two axioms for a linear map: $\...
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2answers
41 views

Matrix/vector proof

Let $C$ be an $m\times n$ matrix with real entries. For vectors $u,v\in\mathbb R^m$, we define $u\sim v$ if there exists a vector $x\in\mathbb R^n$ such that $Cx = u−v$. Prove that for all $u, v,w \in\...
1
vote
1answer
26 views

Proving the images of a function equals specified intervals

I have the following math question: Find the images of each of the following functions: a) $f : [0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = 1/(1 + x^2)$ for $x \geq 0$, b) $...
0
votes
1answer
101 views

Prove that if $ab < 0$ then the equation $ax^{3} + bx + c = 0$ has at most three real roots.

Prove that if $ab < 0$ then the equation $ax^{3} + bx + c = 0$ has at most three real roots. I would need verification on the proof below, thanks! Proof: Let $f(x) = ax^{3} + bx + c.$ Assume ...
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vote
0answers
42 views

Verification of Limit Proof for $\lim_{x \to 9} \sqrt{x-5} = 2$

I am trying to solve this question $\lim_{x \to 9} \sqrt{x-5} = 2$. If you could tell me if I am going about this the right way I'd appreciate it. Given $\epsilon > 0$, Let $\delta =min(9,\epsilon)...
2
votes
0answers
32 views

Every well-ordered set is isomorphic to a unique ordinal number

Every well-ordered set is isomorphic to a unique ordinal number. This is a well-known and important theorem, so i would like to give it a shot by myself. Does my proof look fine, or contain logical ...