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

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2answers
25 views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
1
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2answers
58 views

Find the sum of all 4-digit numbers formed by using digits $0, 2, 3, 5$ - possible formula for competitive exam

Find the sum of all 4-digit numbers formed by using digits 0, 2, 3, 5 without repetition There is a similar question in this site and Eric Tressler has provided a clear method to solve such ...
1
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1answer
73 views

Prove the formula is a contadiction

For every $A$ in Propositional calculus and for every $\rho$ we define: $$A^\rho = \begin{cases} A & \text{if $[|A|]_\rho = true$} \\ \lnot A & \text{if $[|A|]_\rho = false$} ...
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0answers
10 views

PDE $Ru_{zzz}+u_{zzw}=0$: Solution verifcation

I tried to solve this PDE $$Ru_{zzz}+u_{zzw}=0$$ First i substituted $u_{zz}=v$: $$Rv_{z}+v_{w}=0$$ I solved this by the method of characteristics and integrated the result with respect to $z$ ...
1
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1answer
36 views

Proving the column-row expansion of two matrices $A$ and $B$ is equal to the product $AB$

I'm fairly new to proofs with matrices, so determining when I'm allowed to use certain indices to denote arbitrary entries in various matrices is still a challenge. For this problem, I want to prove ...
4
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3answers
100 views

Property of odd ordered elements of a Group

I'm (slowly) working my way through "Abstract Algebra" by Dummit and Foote. In the first set of exercises on group theory, the following question is posed: "Let $G$ be a finite group and let $x$ be ...
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1answer
38 views

Real Analysis Folland Proposition 1.13

Proposition 1.13 - If $\mu_0$ is a premeasure on $\mathcal{A}$ and $\mu^*$ is defined by (1.12) then a.) $\mu^*|\mathcal{A} = \mu_0$ b.) every set in $\mathcal{A}$ is $\mu^*$-measurable ...
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6answers
83 views

Prove by induction the particular inequality $\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$

$\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$ Not sure where I'm going wrong in my Algebra, but I assume it's because I'm adding an extra term. Is the extra term ...
1
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1answer
51 views

Is this family of functions uniformly integrable over $[0,1]$

Let $\mathcal F$ be a family of functions on $[0,1]$ each of which is integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all $[a,b] \subseteq [0,1]$. Is $\mathcal F$ uniformly integrable over $[...
0
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2answers
70 views

Proof verification Show that if $A \subset B$ , then $\overline{A} \subset \overline{B} $(Adherent set ).

Show that if $A \subset B$ , then $\overline{A} \subset \overline{B}.$ $\overline{E}$ = The set of adherent points $x \in \textbf{R}$ such that $ \ \forall \epsilon>0 ,\ (x-\epsilon, x+ \...
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0answers
12 views

Find correlation function and variance of stationary process if we have spectral density

Spectral density of given stationary process is $f(\lambda)=\begin{cases} \frac{\sigma^2}{2\pi},&|\lambda| \leq \lambda_{0}\\ 0,&|\lambda| \geq \lambda_{0}\\ \end{cases}$. Find correlation ...
1
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1answer
30 views

Differentiated series of a power series has the same radius of convergence

I am trying to prove that the radius of convergence of a power series does not change after differentiating term by term. Let $\sum a_nx^n$ be a power series with radius of convergence $R$. Let $R_2$ ...
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2answers
34 views

Proving $\limsup_{n\to\infty}x_n=\liminf_{n\to\infty}x_n$ for convergent $\{x_n\}_{n≥0}\subset\mathbb{R}$

Assuming that for $\{x_n\}_{n≥0}\subset\mathbb{R}$ we have $x=\lim_{n\to\infty}x_n$ I want to show that $x=\limsup_{n\to\infty}x_n=\liminf_{n\to\infty}x_n$. Is the following proof correct? Let $M_n=\{...
3
votes
2answers
67 views

Why do we need to prove a fraction can always be written in lowest terms?

I'm currently reading the notes of a preliminary Math course. Section 3.1.1 contains some proofs using the Well Ordering Principle. One of them is about the always apparent possibility to write a ...
0
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2answers
65 views

Is this proof that $e$ is irrational correct?

I should mention that I still haven't taken Calculus or even Pre-Calculus, which is why I want to ask this. I've seen proofs $e$ is irrational, but not this one. Is this correct, and if it isn't, why ...
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0answers
36 views

Which is finer, co-countable topology or usual topology on $\mathbb{R}$?

We know that the usual topology is finer than co-finite topology on $\mathbb{R}$ How to show the usual topology is finer than co-finite topology on $\mathbb{R}$ And co-countable topology is (in ...
2
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2answers
45 views

Proof verification : limit of the sequence.

im taking my first course in real analysis this summer and I would like some feedback on proof writing. Thank you. Let $\{x_n\}_n$ be a sequence of real number such that $x_n>0$ for all $n \...
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0answers
68 views

The ring $R=C(\mathbb R)$ of continuous functions $f:\mathbb R\to\mathbb C$

Let $R=C(\mathbb R)$ be the ring of continuous functions $f:\mathbb R\to\mathbb C$ where the addition and the product is pointwise defined. Let $$\mathbb m_a=\{f\in R\ |\ f(a)=0\}$$ be a maximal ...
1
vote
1answer
45 views

Verification of change of basis calculation

A short question just to check regarding a change of basis: Let $$A = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \end{bmatrix} \right\}$$ be a basis ...
3
votes
2answers
62 views

Show that $|h(x)|\le\frac{x^2}2 M$ if $|h''(x)|\le M$ and $h(0)=h'(0)=0$ on $[0,a]$

This is the exercise 5.3.6 of Understanding analysis 2ed by Abbott. Show that $|h(x)|\le\frac{x^2}2 M,\forall x\in[0,a]$ if $|h''(x)|\le M,\forall x\in[0,a]$ and $h(0)=h'(0)=0$ on $[0,a]$ It is ...
1
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2answers
46 views

Why do the solids of revolution of $y=x$ about the OX and OY axes have different volumes?

Given a line $y=f(x)$, the formulae for the solids of revolution are: $$V_{OX} = \pi \int_{a}^{b} f(x)^{2} \mathrm{d} x$$ $$V_{OY} = 2\pi \int_{a}^{b} x f(x) \mathrm{d} x$$ Applying these formulae to ...
3
votes
0answers
72 views

Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?

I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous. Statement:...
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0answers
27 views

$f_n\rightharpoonup f$ weakly in $\sigma(L^1,L^{\infty})$ and $\|f_n\|_1\to \|f\|_1$ no implies $\|f_n-f\|_1\nrightarrow 0$ [duplicate]

I'm studying Brezis' book of Functional Analysis. I'm trying to do the exercise $4.19$ and I would like of a little help. Le be $\Omega=(0,1)\in \Bbb{R}.$ Construct $(f_n)$ in $L^1 (\Omega),$ $f_n ...
4
votes
2answers
89 views

Proof checking : Show that $\bigcap\limits_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right)= \{0\}. $

Can anyone check my proof please? Thank you. Show that $$\bigcap\limits_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right)= \{0\}.$$ Let take an arbitrary $x \in \textbf{R}$ such that $|x|\...
3
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4answers
86 views

Proof writing: $\sum_{n=1}^{\infty}| a_n|<\infty $ implies $\sum_{n=1}^{\infty} a_n^2<\infty $.

Let $\sum_{n=1}^{\infty} a_n $ be an absolutely converging series. By definition, this means $\sum_{n=1}^{\infty} \lvert a_n\rvert $ converges. We want to show that $\sum_{n=1}^{\infty} a^2_n $ ...
2
votes
2answers
51 views

If $A$ is a matrix with a row of zeros, then the product $AB$ also has a row of zeros

I know there is a similar question posted on this site, but it does not have a proof, and I would rather have my proof for the problem criticized than look at a proof as an answer on the other post. ...
3
votes
2answers
31 views

Verification of proof for when a number is divisible by 4

I have never taken a number theory course and so am only going off of the first few chapters in an introductory number theory book. The divisibility property I wish to prove is the following: Define ...
2
votes
1answer
66 views

Topology and Borel sets of extended real line

Let $\mathcal{B}_{X}$ denote the Borel $\sigma$-algebra on $X$. I'm reading a book on real analysis by Folland and he defines $$\mathcal{B}_{\overline{\mathbb{R}}} = \{ E \mid E \cap \mathbb{R} \in \...
1
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0answers
10 views

$||D_hu||_{L^2(Q_+)} \leq ||\nabla u||_{L^2(Q_+)}$ for $u \in H_0^1(Q_+)$

I want to show the following statement: Given $u \in H^1_0(Q_+)$ with $supp(u) \subset \{x \in \mathbb{R}^n\ | \ (\sum_{i=1}^{n-1} |x_i|^2)^{1/2}< 1 - \delta, \ 0 \leq x_n < 1-\delta \}$ y $...
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0answers
12 views

If $f$ is differentiable on an interval $A$ containing zero with $(x_n)\to 0$ and $x_n\neq 0,\forall n\in\Bbb N$ show that…

Im interested in the part b) of the question. If $f$ is differentiable on an interval $A$ containing zero with $(x_n)\to 0$ and $x_n\neq 0,\forall n\in\Bbb N$ show that... a) If $f(x_n)=0,\...
2
votes
2answers
56 views

Basic question about ideals in a polynomial ring.

I would be very grateful if someone would verify or refute the following solution. Many thanks! Q) Find infinitely many distinct ideals of $\mathbb{C}[X,Y]$ which contain the principal ideal $(X^3-...
0
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0answers
20 views

Let $F=K(u)$ where $u$ is transcendental over $K$, prove that it is algebraic over $E$, where $K \subset E \subseteq F$

Let $F=K(u)$ where $u$ is transcendental over $K$. Prove that it is algebraic over $E$, where $K \subset E \subseteq F$. The method I tried for the above question was as follows: Choose $v \in E/K$ ...
0
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0answers
29 views

Solving a system of differential equations with a repeated eigenvalue

$\vec{x}' = \begin{bmatrix}4&-2\\8&-4\end{bmatrix}\vec{x},$ I'm getting $\lambda = 0$ as an eigenvalue And the resulting eigenvector $\vec{v} = \begin{bmatrix}1\\2\end{bmatrix}$ I ...
5
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1answer
53 views

Prove if $A ⊆ B$, then $A ∩ C ⊆ B ∩ C$.

For any sets $A, B$, and $C$ Assume $A ⊆ B$, and suppose, $x\in (A∩ C)$. Then $x\in A $ and $x\in C$ by definition of $A ∩ C$. Since $A ⊆ B$ it follows that if $x\in A$ then $x\in B$. Thus, ...
1
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1answer
26 views

Prove by induction $\frac{7}{8}+7\left(\frac{1}{8}\right)^2+…+7\left(\frac{1}{8}\right)^n=1-\frac{1}{8^n}$ for every $n \in \mathbb N$

$\frac{7}{8}+7\left(\frac{1}{8}\right)^2+...+7\left(\frac{1}{8}\right)^n=1-\frac{1}{8^n}$ for every $n \in \mathbb N$ I first set $n=k$: $\frac{7}{8}+7\left(\frac{1}{8}\right)^2+...+7\left(\frac{1}{...
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3answers
29 views

Proof that $Qo(n) = 2(\sum_{i=1}^{n-1}i)+2n = n^2 + n$

So i would appreciate if someone explain to me the step by step on how do i get this result $Qo(n) = 2(\sum_{i=1}^{n-1}i)+2n = n^2 + n$ How do you proof that it is $=n^2+n$ ?
1
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1answer
34 views

Notation matter concerning 'or'-elimination

I have to show that $\{(\phi\lor\psi),(\lnot\phi)\}\vdash\psi$ using the following natural deduction rule: I don't know which of these is correct in term of notation: Could you please tell me? ...
2
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1answer
15 views

Bounding the number of points at infinity of a curve of degree $n$.

I'm trying to prove the following statement. Let $C$ be a curve of degree $n$. Give a bound for the number of points at infinity. I tried it for $C$ defined by a polynomial in two variables only. ...
1
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2answers
22 views

Any T1 space having an isolated point is disconnected

There's an example in Willard's General Topology that says 26.2 b) Any $T_1$ space having an isolated point is disconnected. The proof I made is: Lets assume that $X$ is $T_1$ with more ...
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0answers
28 views

Let $f$ be a family of sets. Prove that there is a unique set $A$ such that $f \subseteq \mathscr P (A)$.

Suppose $f$ is a family of sets. Prove that there is a unique set $A$ such that $$f \subseteq \mathscr P (A)$$. Proof: Let $A= \cup f$. Let $X \in f$, and now let $y \in X$. Clearly, $y \in \cup f$...
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1answer
22 views

Find an equation of the tangent plane to the given surface

Question: Find the equation of the tangent plane to the surface with equation $z = 3y^2-2x^2+x$ at the point $(2,-1,-3)$. My attempts: $\nabla f_x$ $= -4x+1$ $\nabla f_y$ $= 6y$ Setting up the ...
3
votes
2answers
86 views

Is this logic of solving $\frac{0}{0}$ correct [duplicate]

I have seen such proofs many times and was unable to prove where it was wrong (not a math person-my bad). For example the following proof for $\frac{0}{0} = 2$ looks like it is proven correctly - but ...
0
votes
1answer
40 views

Extrema of $f(x)=g(|x-a|^2,|x-b|^2,|x-c|^2):\Bbb{R}^n\to \Bbb{R}$ in $S=\{x: |x|=1\}\subset \Bbb{R}^n$ is a linear combination of $a,b,c$

Since I am getting pretty close to the final exams, I would really yield from having my practice challenged and corrected. Question: Let $a,b,c\in \Bbb{R}^n$ be independent vectors, and $g\in C^{1}(\...
0
votes
3answers
51 views

Given ΔABC with sides a,b,c and circumradius R. Prove this…

Given $\triangle ABC$ with sides $a,b,c$ and circumradius $R$. Prove that $$\cot A+ \cot B +\cot C= \frac{a^2+b^2+c^2}{abc}R$$ I have no idea what to do.
1
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0answers
113 views

Green eyes/Common Knowledge problem proof verification

I was trying to solve the common knowledge problem, but am not sure if my proof is accurate. Here is a rough statement of the problem : 'An island consists of $k$ people with green eyes, all ...
1
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1answer
35 views

Proving $n < 2^n$ by Cantor's theorem

So we know Cantor's Theorem is of course. For any set $S$, the power set $P(S)$ has a strictly greater cardinality, $\iff \#S < \#P(S)$. We seek to prove $n < 2^n$ using this information. I ...
2
votes
0answers
30 views

Prove $r \circ f = s \circ f \implies r =s$ holds for a surjection $f$.

Frankly I'm not convinced what I have is technically correct so I was hoping to get some verification. Suppose we have functions $f:A\rightarrow B$, $r:B\rightarrow C$, and $s:B\rightarrow C$. ...
4
votes
3answers
61 views

Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$

I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not. This is probably off-base, but it's what ...
1
vote
1answer
62 views

$\varepsilon - \delta$ proof that $f(x) = x^2 - 2$ is continuous - question concerning the initial choice of $\delta$

I just realized I did non really internalized $\varepsilon - \delta$ proofs. Here there is an attempt, with some general questions I have. Proposition: $f(x) := x^2 - 2$ is continuous. ...
3
votes
3answers
44 views

Prove: if $f:\mathbb{N} \rightarrow A, g:\mathbb{N} \rightarrow B$ are surjections, there exists a surjection $h:\mathbb{N} \rightarrow A \cup B$

I chose my own sets here for A and B as countably infinite pairwise disjoint subsets of $\mathbb{N}$. Can I do this with finite subsets and get an easier answer with the same result? Suppose $A = \...