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

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0
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2answers
108 views

Artinian rings have finite length

In a recent question of mine here I asked whether it is true or not that Artinian (commutative) rings have finite length. I came up with a proof, and I want to know if it is valid. So, I want to ...
0
votes
1answer
35 views

Is my proof of the weakening principle correct?

Could you please check if my proof of the weakening principle of intuitionsitic logic is correct? $$\Gamma \vdash B \Rightarrow \Gamma, A \vdash B$$ Proof: Let $\Gamma \vdash B$. Hence there is a ...
7
votes
4answers
212 views

Looking for a direct proof of the following exercise

A friend of mine told me about the following problem: Let $\{r_n\}$ be a sequence of rational numbers such that $\lim_{n\to\infty}r_n=x\in\Bbb R,$ $r_n\neq x,$ for every $n\in\Bbb N$ and ...
1
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2answers
64 views

Try to prove a generalization of the IVT?

Here is the statement : "Let $f: (a,b) \to \mathbb{R}$ a continuous function on $(a,b)$ with $a<b$ and $a,b \in \bar{\mathbb{R}}$. Then for all $u \in \left(\lim \limits_{x\to a} f(x), \lim ...
2
votes
1answer
43 views

Trouble with Indexing in Linear Algebra Proof

I just started going through Jim Hefferon's book on Linear Algebra, and I'm having trouble with one of the proofs. I understand its overall structure but some of the indexing in the proof seems off by ...
-4
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2answers
124 views

Looking for a verification or refutation my attempted proof of why the Collatz conjecture is probably false. [closed]

Most people think that the Collatz conjecture is true, but I think that I can prove the opposite. Let's make two functions, $f(x)$ and $g(x)$. $f(x) = $ The amount of numbers that can be solved in x ...
0
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1answer
36 views

Prove that $\sum_{k=1}^{n-1}k^{3}\leq \frac{n^{4}}{4}\leq \sum_{k=1}^{n}k^{3}$ for all $n\geq 2$.

Prove that $$\sum_{k=1}^{n-1}k^{3}\leq \frac{n^{4}}{4}\leq \sum_{k=1}^{n}k^{3}$$ for all $n\geq 2$. This is just a random exercise to improve my proof techniques. I want to show it by induction ...
2
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1answer
28 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
6
votes
2answers
254 views

Is this limit proof correct?

I am currently studying the formal defenition of the limit. One of the examples given by my book is the following: Prove that: $$ \lim_{x \to 3} x^2 = 9 $$ So, using only the defenition of the limit, ...
0
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2answers
42 views

A sufficient condition for a set to be dense.

A subset $A$ of a topological space $(X,\tau)$ is said to be dense if $\overline A=X$. Prove that if for each open set $O\neq\varnothing$ we have $A\cap O\neq\varnothing$, then $A$ is dense in $X$. ...
1
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1answer
22 views

Proof Verification Regarding Uniform Continuity

Assume that $g$ is defined on an open interval $(a, c)$ and it is known to be uniformly continuous on $(a, b]$ and $[b, c)$, where $a < b < c$. Prove that g is uniformly continuous on $(a, c)$. ...
1
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1answer
41 views

Showing equivalence of seminorms

Let $K =[0,1]$ and let $X \subset C^{\infty}(K)$ be the subspace of all functions vanishing on the end points of $K$. Show that the following seminorms are equivalent: $||D^nf||_1$ $||D^nf||_2$ ...
4
votes
3answers
63 views

Show that $\frac{1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1)}{1\cdot 2\cdot 3\cdot \ldots \cdot n}\leq 2^{n}$ for all $n\in\mathbb{N}$.

Show that $$\frac{1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1)}{1\cdot 2\cdot 3\cdot \ldots \cdot n}\leq 2^{n}\qquad (n\in \mathbb{N}).$$ I want to show the last step, that is, the inductive step. ...
1
vote
1answer
41 views

$(X_n)_{n\in\mathbb{N}}$ independent Cauchy-distributed random variables. Convergence of $n^{-\gamma}(X_1+\cdots+X_n)$

I want to solve the following exercise but i am unsure if my ideas are correct or not. Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables with probability density $$ ...
2
votes
1answer
52 views

Some questions about Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras

I have some questions about Joachim Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras, which is found in this paper. For this post, let us adopt the ...
2
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2answers
35 views

Proofs: Induction on Handsakes

Here is the problem: Suppose $n$ people are at a party, and some number of them shake hands. At the end of the party, each guest $G_i$, $1 \leq i \leq n$ shares that they shook hands $x_i$ times. ...
4
votes
1answer
209 views

How do I prove that there is no other :$k=9,12,18$ for which this fails :$\sigma^k(114) \equiv 0\mod 6 $?

let $\sigma(n)$ be the sum of divisors for a positive integer for example : $$\sigma(6)=1+2+3+6=12$$ . I have performed some calculations in wolfram alpha about the sum divisors of this number: ...
0
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0answers
30 views

Number Theory by Andreescu and Andrica Problem 1.6.2 Solution

Problem 1.6.2 in Number Theory by Andreescu and Andrica is taken from 1997 Czech and Slovak Mathematical Olympiad, and is stated as follows: "Show that there exists an increasing sequence $\{a_n\}$ ...
2
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1answer
38 views

Understanding Apostol's proof of the Lagrange remainder theorem

The proof is: What I don't understand in this proof is (1): how does he manage to bring about a third subtraction term in the second last expression. I assume that he is expressing the function ...
3
votes
1answer
42 views

“Counterexample” to this characterization of lim sup?

I came across an exercise (Exercise 10, Ch. 1, Marsden's elementary classical analysis, 2nd ed.) that gives a characterization of lim sup I had never seen, which can be rephrased as follows: Let ...
7
votes
2answers
273 views

Was Smullyan really wrong?

EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay! There was a question asked on Puzzling recently, titled ...
1
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0answers
16 views

To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ ( Riemann Integration)

To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ Question : Let $[a,b] \subseteq R$ be a non degenerative closed bounded interval and let $f,g :[a,b] \rightarrow R$ be functions .Suppose that there is ...
2
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0answers
23 views

Is $L^p(\Omega) = L^p(\bar {\Omega})$?

Is it true that $L^p(\Omega) = L^p(\bar {\Omega})$, where let us say $\Omega$ is a bounded domain of $\mathbb{R}^n$ with smooth boundary? I think it is true, because $\partial \bar {\Omega}$ has ...
0
votes
2answers
85 views

Proof that $\lim_{x\to 0} \sin(1/x)$ does not exist using contradiction

I am currently working through Apostol's Calculus, and I was hoping that someone could verify that the proof that I wrote for one of the problems actually proves the assertion. Prove that ...
2
votes
1answer
25 views

$X$ connected in the order topology $\Rightarrow$ every $A\subset X$ that has an upper bound has a supremum

I have to prove $X$ connected in the order topology (w.r.t. a linear order <) $\Rightarrow$ every $A\subset X$ that has an upper bound has a supremum My attempt: Reason by contradiction: ...
0
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3answers
28 views

Let $H$ be a non-zero subspace of $V$, and let $T(H)$ be the set of images of vectors in $H$. Prove that $\dim(T(H))\leq \dim(H)$.

Let $V$ and $W$ be finite-dimensional vector spaces and $T$ be a linear transformation $T:V\to W$. Let $H$ be a non-zero subspace of $V$, and let $T(H)$ be the set of images of vectors in $H$. ...
1
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0answers
31 views

Proof Verification: Converse of Intermediate Value Theorem

A function $f$ is increasing on $A$ if $f(x)\leq f(y)$ for all $x<y$ in $A$. Show that the Intermediate Value Theorem does have a converse if we assume $f$ is increasing on $[a,b]$. The converse ...
0
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0answers
36 views

For what functions is this theorem correct?

Theorem$_0$: If $g:\mathbb{C}^k \to \mathbb{C}^k$ sends $(t a_1,...,t a_k)$ to $(t^{\alpha_1}b_1,...,t^{\alpha_k}b_k)$ for all $t\in \mathbb{C}$, the preimage of any point has size $\alpha_1 \cdots ...
0
votes
1answer
24 views

Filters and filter bases in order theory

Hi I would like to confirm the following ideas regarding filters in order theory: By definition I have that a filter is a subset of a poset $(P, \leq)$ which satisfies: $\mathcal{F}$ is non-empty. ...
0
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4answers
46 views

Calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$

I'm trying to calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$. I'm really bad with groups so I'd appreciate a verification of my conclusion: If $m$ is even then $\forall x\in \mathbb Z_m$ we get ...
3
votes
1answer
125 views

Proof verification of a weak version of Bezout's Theorem

I'd like to make sure here that my reasoning seems sound. I am working from Kirwan's book on algebraic curves. I was not totally happy with her proof of this theorem, so I wanted to see if I could ...
0
votes
2answers
150 views

Is this induction proof of the Fundamental Theorem of Algebra rigorous?

I was trying to find a suitable proof of the Fundamental Theorem of Algebra at an undergraduate level which avoided abstract linear algebra, as I have not yet begun it. However, I came across this ...
2
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3answers
178 views

How can I know if my conjecture is not lacking mathematical formality? [closed]

I'm a teenager and student who came up with his own conjecture. Because I'm not a mathematician and I haven't got the knowledge yet, I would like to know if my conjecture doesn't exceed the limits of ...
2
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2answers
55 views

The topological space $\left(X,2^X\right)$ is metrizable

Prove that for each set $X$, the topological space $\left(X,2^X\right)$ is metrizable, where $2^X$ is the power set. What I'm not sure is what are the conditions for a topological space to be ...
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0answers
27 views

trace calculation of an operator valued matrix

Heyho, i've got problems understanding a certain calculation of the trace of an operator valued matrix right now. We've got the Matrix $T(\lambda)= \begin{pmatrix} A(\lambda) && B(\lambda) ...
2
votes
1answer
46 views

How to Prove Plancherel's Formula?

I have difficulty in proving Plancherel's formula in Fourier transform. Here is what I have thought: In this question, I denote complex conjugation by an overline and (inverse) Fourier transform is ...
5
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2answers
65 views

Accumulation points of $\{ 2^{-n} + 5^{-m} : n,m \geq 1 \}$

Determine all of the accumulation points of the following sets in $\mathbb{R}^1$ and decide whether the sets are open or closed or neither. The set $X$ of all numbers of the form $2^{-n} + ...
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0answers
48 views

Is this $\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $ alternating series for all values of $\theta$?

I have tried to do other form of alternating series I got this: $$\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $$ Can I say that the above series is alternating series for all ...
1
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1answer
27 views

Ways to stack 65 different disks in 3 piles with constraints.

How many ways are there to stack 65 different disks in 3 piles if pile 1 but have at least 15 disks and pile 3 must be non-empty. Attempt: 1) Ways to arrange all the disks in a horizontal line: ...
1
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0answers
41 views

How do we obtain this inequality? A question concerning an argument in Stroock and Varadhan 1971

The problem comes from the article of Stroock and Varadhan [diffusion processes with boundary conditions (1971) ]. So far I have followed, but in the next page I got lost: I don't follow the ...
2
votes
1answer
35 views

Graph of a non-continuous function is closed

Exercise. Let $\ f\colon\mathbf R\to\mathbf R$ be defined by $$f(x)=\begin{cases}\frac{1}{x},\ x>0\\0,\ x\leq 0\end{cases}$$ Prove that the graph $\Gamma_f:=\{(a,f(a)):a\in\mathbf R\}$ of ...
1
vote
2answers
37 views

Squaring both sides of an inequality: attempt to prove a general rule

I have attempted to produce a proof of the intuitive rule for squaring inequalities, according to which, given any two numbers x and y and regardless of their sign, 1) if |x| < |y| then ...
1
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1answer
50 views

Topology on partially ordered set

Let $(X, \leq )$ be a partially ordered set. How would you define a topology on $X$ such that the closed sets are precisely the order-closed sets? Where $B \subset X$ is order-closed if ...
0
votes
1answer
33 views

Question concerning a proof on Stroock and Varadhan 1971

In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads: where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ ...
3
votes
0answers
61 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
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0answers
21 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
1
vote
3answers
39 views

Showing that a subgroup of an abelian group is normal—is this sufficient?

When asked to show that a subgroup $H$ of the abelian group $G$ is normal, does it suffice to say: first, $H$ is a subgroup, so it contains the identity element of $G$ and inverses $h^\prime$ for ...
0
votes
2answers
17 views

Replacing occurrences of the same integer as follows: is it legitimate in subsequent steps of a proof?

I need to spot the incorrect step in a spoof (false proof) and the first two lines are: $-5 = -5$ $-7+2 = -4-1$ Of course both the right- and left-hand side of the equation in the second line are ...
3
votes
1answer
52 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
1
vote
1answer
41 views

How many surjective functions $f: X \to \{1,…,j\}$?

How many surjective functions $f: X \to \{1,...,j\}, |X|=j \cdot k.$ can be defined if they must satisfy: $$ |\{x\in X: f(x)=r\}|=|\{x\in X: f(x)=s\} \forall r,s\in \{1,...,j\} $$ My attempt: From ...