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

learn more… | top users | synonyms

1
vote
2answers
14 views

A finite set and the set of its fixed points under any involution have cardinalities of the same parity

I am trying to write down a formal proof of the following fact: Let $A$ be a non-empty finite set and $f$ an involution on $A$. If $A'$ is the set of fixed points of the involution $f$, then $|A| ...
2
votes
3answers
64 views

Prove that $\sum_{d|n}\phi(d)=n$ where $\phi$ is the Euler's phi function, $n,c\in\mathbb{N}$

Here is a very elementary number theory proof using strong induction. Please mark/grade. Prove that $$\sum_{d|n}\phi(d)=n$$where $\phi$ is the Euler's phi function, $n,d\in\mathbb{N}$ First, ...
1
vote
2answers
45 views

Proving that any connected graph has a vertex whose removal results in a connected graph

I want to prove that: for any simple, connected graph there is at least one node whose removal results in a connected graph. Here is my proof: Suppose that a graph $G$ is simple connected graph with ...
1
vote
2answers
64 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
3
votes
1answer
28 views

Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
1
vote
1answer
23 views

Asymptote criterion

Let $f:(a, \infty)\to \Bbb R$ be a differentiable function such that exists $\lim_{x\to\infty}f(x)=l<\infty$ and exists (in the sense it can also be infinity) $\lim_{x\to\infty}f'(x)$. Under these ...
3
votes
1answer
27 views

The set is closed (resp. open) iff the complement set is open (resp. closed)

There's a theorem in my small danish course book. Let $(M,d)$ be a metric space. Theorem: The concepts of open and closed are dual: A set $A\subseteq M$ is closed (resp. open) if and only if the ...
0
votes
1answer
32 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
0
votes
2answers
58 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
2
votes
1answer
22 views

Number of square matrices of order $n$ where each row and each column has at most one $1$

What is the number of square matrices of order $n$ with the property that each row and each column has at most one $1$, and $0$s elsewhere? For example, when $n=2$, there are $7$ such matrices: ...
0
votes
1answer
25 views

Proving B Congruent C given AB congruent AC

This is a very trivial question, i seem to have arrived at a proof for an excercise but the proof just doesn't feel.. right. It is too small and simple. The fact to be proved is that if $AB\equiv AC$ ...
0
votes
0answers
33 views

Separated subsets of $\mathbb{R}^k$

Let $A$ and $B$ be separated subsets of some $\mathbb{R}^n$, suppose $a\in A, b\in B,$ and define $p(t)=(1-t)a+tb$ for $t\in \mathbb{R}^1$. Put $A_0=p^{-1}(A), B_0=p^{-1}(B)$. (a) Prove that $A_0$ ...
0
votes
0answers
18 views

Isomorphic Fields Leading to Isomorphic Splitting Fields

Link to Original Text: Theorem 10.6 Let $F, F'$ be two fields isomorphic via $\varphi$. Suppose that $f = \sum_{i=0}^m c_ix^i \in F[x]$ splits in $E$, and that the corresponding $f' = \sum_{i=0}^m ...
2
votes
1answer
50 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
1
vote
1answer
45 views

A proof that every set of natural numbers contains a minimal element

I'm currently trying to extend my basic knowledge and in order to do so, I started with the Peano-axioms. I think, I understand the underlying thoughts and I want to prove the following theorem using ...
1
vote
1answer
63 views

What is wrong with my brute-force approach to proving that $\mathbb R$ as a metric space obeys the triangle inequality?

In a self-study of metric spaces, I'm looking at the very basic exercise of proving that $(\mathbb R, |y-x|)$ is a metric space. The sticking point was the triangle inequality. I did manage to ...
2
votes
1answer
59 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
0
votes
2answers
74 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
1
vote
0answers
17 views

Proof Check on Change of Variables Result

Let $g: I \to \mathbb{R}$ be strictly increasing with continuous derivative on an open interval $I \subset \mathbb{R}$. Let $\mu$ be the measure on $(I, \mathcal{B}(I))$ with density $g^\prime$ ...
1
vote
1answer
40 views

An exercise on implication (proof and logic)

This question is derived from the book "How to think like a Mathematician" which does not have solutions to its questions. Following exercise is on implications: Suppose that students were told that ...
2
votes
5answers
56 views

Is the substitution of standard angles while proving the equality of trigonometric formulas allowed?

Here is a problem that my class 10 maths teacher gave me: Prove that $\sec^4\theta$ - $\sec^2\theta$ = $\tan^4\theta$ + $\tan^2\theta$ She expected me to use trigonometric identities to prove ...
0
votes
0answers
27 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
3
votes
2answers
39 views

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$ I am not sure that whether the way that i prove this statement is correct. Since $[G:H]=2, \forall a\in G,G/H=\{H,Ha\}$ Hence $Ha^2=H ...
0
votes
2answers
23 views

Define a relation $D_n$ on $S$ by $xD_ny$ if and only if $x\mid y$. Determine if it's a poset.

Here is the question I am currently working on (screenshot): I'd appreciate some suggestions/guidance for part (a), proving that $D_n$ is a partial order. Reflexive: Let $x \in \mathbb{Z}$ ...
2
votes
1answer
59 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
1
vote
0answers
28 views

Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$? Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of ...
1
vote
3answers
100 views

Proof of the Union of Power Sets

I saw the proof at ProofWiki. I know well that $X\in \mathcal{P}(A)\iff X\subseteq A$ where $A$ is a set, but why can't I show the following way \begin{align} X\in (\mathcal{P}(S)\cup ...
1
vote
1answer
18 views

Generalization of the Saccheri-Legendre Theorem Proof

So I'm working on generalizing the Saccheri-Legendre Theorem to convex $n$-gons. $\underline{\text{Saccheri-Legendre Theorem:}}$ The sum of the angles of a triangle is at most $180^\circ$. A ...
4
votes
3answers
191 views

Baby Rudin claim: $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}…$ converges

This sequence is a rearrangement of the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}...$. Note that at this point in the text we do not have any theorem about the convergence of ...
0
votes
1answer
41 views

Is the following statement true? if so prove. ∃ a,b ∈ R, ∀x ∈ R, (ax+b=x)

Is the following statement true? if so prove. $(\exists \, a,b \in \mathbb{R}), (\forall x \in \mathbb{R}), (ax+b=x)$ Having a hard time proving this statement.
0
votes
1answer
36 views

Show that $A\subseteq B\implies A^{\circ} \subseteq B^{\circ}$ in a different way.

Let $A$ and $B$ be subsets of a metric space $(M,d)$. If $A\subseteq B$, then $A^{\circ} \subseteq B^{\circ}$. Proof : Assume that $a\in A^{\circ}$. Then there exists a $r>0$ such that ...
1
vote
0answers
35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
1
vote
1answer
69 views

Category of Sets w/ 17 Elements: There does not exist a direct product? (Lots of questions here)

I'm having a pretty hard time with this. I'm asked to show that, in the category of sets with exactly 17 elements, no two objects have either a direct product nor direct sum. Part of me doesn't even ...
1
vote
1answer
48 views

Is every image of a loop in Hausdorff space, homeomorphic to $S^1$?

Let $S^1$ be the 1-sphere with respect to the standard norm on $\mathbb{R}^2$. Let $X$ be a Hausdorff space and $\alpha:[0,1]\rightarrow X$ be a loop. Define $C=\alpha([0,1])$. Define ...
0
votes
2answers
54 views

Induction question regarding Universe

I was given a question that looks like this. Prove that for each $2 \le n\in \mathbb N$, if $X_1,\ldots,X_n$ are subsets of some universe $U$, then the following is true: $$(X_1 \cup\cdots\cup ...
0
votes
4answers
73 views

Prove for each $n\in \mathbb{N}, 1^3 + 2^3 +\cdots+ n^3 = \frac{n^2(n+1)^2}{​4} ​ ​​​$ [duplicate]

So I was given a proof by induction question and here is my attempt $$1^3 + 2^3 + 3^3+\cdots+n^3= \frac{n^2(n+1)^2}{4}$$ $n=1$ $1=1$ Induction step: Assume statement is true for $n=k$, show true ...
0
votes
1answer
55 views

Equivalent conditions of Lebesgue measurable sets

Hi I'd appreciate if someone can check the following exercise any suggestions are welcome. Thanks ;) Let $A$ a subset of ${\bf{R}}^d$ show that the following conditions are equivalent: (i) $A$ ...
1
vote
1answer
13 views

Prove that an underdetermined system of cannot have a unique solution(Is this proof correct?)

I know I misspelled underdetermine but is this proof correct? How can I improve it either way? Side Remark: Anyone who is down-voting please can you understand I new to this site and somewhat ...
0
votes
1answer
25 views

Laplace equation in spherical coordinates

I am trying to calculate the Laplace equation($\Delta f =\partial_x\partial_xf + \partial_y \partial_y f + \partial_z \partial_z f = 0$ ) in $\Bbb{R}^3$ for spherical coordinates. $$g(r, ...
0
votes
3answers
57 views

proof by induction $2^n \leq 2^{n+1}-2^{n−1}-1$

My question is prove by induction for all $n\in\mathbb{N}$, $2^n \leq 2 ^{n+1}-2^{n−1}-1$ My proof $1+2+3+4+....+2^n \leq 2^{n+1}-2^{n−1}-1$ Assume $n=1$,$1 ≤ 2$ Induction step Assume statement ...
4
votes
1answer
61 views

If $h(x)=f(g(f(x)))$ is bijective, what do we know about $f,g$?

Question: If $h(x)=f(g(f(x)))$ as a function $\mathbb R \rightarrow \mathbb R$ is bijective, what do we know about $f,g$, which are also functions $\mathbb R \rightarrow \mathbb R$? Is my proof ...
0
votes
2answers
22 views

If $W$ admits an injection of $k$-algebras in its coordinate ring, then $W$ is an unirational variety

I'm studying algebraic geometry from "Introduction to algebraic geometry" by Hassett, and I did not understand a step in his proof of the following result (page 52): "If $W$ is an affine variety ...
1
vote
1answer
33 views

Is it true that kernel of an endomorphism $f$ is not null if $0$ is eigenvalue of $f$?

$f$ is an endomorphism of the vector space $V$. Is it true that if $0$ is an eigenvalue of $f$ then $\ker(f)$ contains at least a non-null vector? My attempt: from the definition of eigenvector, ...
1
vote
1answer
38 views

Proof check - infinite-dimensional $\mathfrak{sl}(2, \mathbb{F})$-module

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with the usual basis $(x, \ y, \ h)$ and $\text{char}\,\mathbb{F}=0$. Let $Z(\lambda)$, $\lambda\in\mathbb{F}$ the infinite-dimensional $L$-module spanned by ...
1
vote
0answers
47 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
3
votes
1answer
49 views

Stalks of the sheaf of total quotient rings

Let $X$ be a scheme, for each $U$ open in $X$, let $S(U)$ be the set consisting of elements of $O_X(U)$ whose image in $O_{X,p}$ is a non-zerodivisor for every $p\in U$. In particular, if $U = ...
2
votes
0answers
28 views

Proof-validation of centroid's existence

So, a friend of mine came up with this unorthodox proof of the centroid's existence so I figured I could share it here so that someone can confirm that it's a fine one. I think it is correct, but I ...
1
vote
0answers
16 views

Support of a discrete measure

According to Wikipedia (and some books), a discrete measure on the real line is a measure $\mu$ whose support $\text{supp}(\mu)$ is at most a countable set. This definition seems to be inconsistent ...
4
votes
4answers
1k views

If the square of a number is even, then the number if even. Isn't that not true for 2?

I'll quickly go over my understanding of it: If a number $n^2$ is even, then $n$ is even. The contrapositive is that is that if $n$ is not even (odd), then $n^2$ must also be not be even (be odd). ...
1
vote
1answer
39 views

A polynomial's irreducibility in $\Bbb{Z}_p$

Show that if $f$ is irreducible in $\Bbb{Z}_p[x]$ then $f$ divides $x^{p^n} - x$ for some $n \in N$. I know that: $f$ is irreducible, so $F = \Bbb{Z}_p / {\left\langle f\right\rangle}$ is a ...