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

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0answers
18 views

Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of ...
-2
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0answers
27 views

Using induction to prove for any n≥2 and any sequence prove the following claim. [on hold]

So i was given a question that introduces a mystery function $P:\Bbb N \to \Bbb N$. We don't know a formula for $P$ but we do know that $P$ satisfies the following relationship for all $a_1,a_2 \in ...
1
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1answer
39 views

Injections, Surjections, Bijections [on hold]

So i was given a question that asks me to determine whether the function is injective, bijective, or surjective. If you answer bijective than determine the functions inverse, domain, and target space. ...
2
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2answers
45 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
4
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1answer
90 views

Possible proof of Fermat's Last Theorem for prime exponents greater than 2

I would appreciate if someone could check my attempt in proving the Fermat's Last Theorem for prime exponents greater than $2$. Firstly, let's prove a couple of lemmas which state that sum or ...
0
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2answers
31 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
0
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2answers
22 views

Spot the error in experimenting with contradiction on 5's rationality.

Let $5=\frac ab$ $\forall\ a,b\ \epsilon\ N$. And $(a,b)=1$ Squaring both sides, $25b^2=a^2$ Thus, $25|a^2$; $25|a$ So $a=25m$ Substituting, $25b^2=25^2m^2$ So $b^2=25m^2$ So $25|b$ (By the ...
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0answers
56 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective [on hold]

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
0
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1answer
23 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
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0answers
41 views

prove that for any 2n≥2 and any \a ​1 ​​ ,…,a ​n ​​ ∈N, we have the following: [on hold]

So the question I was given goes like this we will introduce a mystery function,P:N→N. We don't know a formula for P (and we won't be able to determine one!) but we do know that P satisfies the ...
1
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3answers
65 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
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3answers
72 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
0
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3answers
57 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
2
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2answers
22 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
0
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3answers
34 views

Trigonometical identity proof

I was given a proving sum: $\sec(x) + \tan (x) = p$, prove $\frac{p^2-1}{p^2+1} = \sin (x)$ I went head on and tried to directly do it by solving the LHS: $\sec(x) + \tan(x)$ = $\frac{1}{\cos(x)} ...
1
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1answer
28 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
3
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4answers
96 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
0
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0answers
31 views

Proof Check In: Prove that $(\mathbb{Z}_n, +)$, the integers (mod $n$) under addition, is a group.

I received some help and direction on this from some users a few days ago, and have tried to take that information and craft it into something proofy. I would appreciate general suggestions, edits, ...
2
votes
3answers
48 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
0
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1answer
17 views

Composition of analytic function with arithmetic function

Consider an arithmetic function $g$ with codomain $\{a,b\}$ and a function $f$ which is analytic on some domain including $\{a,b\}$. We therefore have $$f(g(n))=\sum_{k=0}^\infty c_k (g(n)-a)^k$$ and ...
2
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1answer
17 views

Assumption on characteristics in an exercise about roots of unity

I'm solving the following exercise: "Let $K$ be a field, $char(K) \nmid 2n$ for $n \geq 1$ an odd integer. If $K$ contains a primitive $n$-th root of unity, then it also contains a primitive $2n$-th ...
2
votes
5answers
76 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
0
votes
1answer
51 views

Prove that $(a,b]\subseteq \mathbb{R}$ is not open.

I want to prove myself that a half-interval $(a,b]\subseteq \mathbb{R}$ is not an open set. I checked it in here. My proof: We wish to prove that $b\notin (a,b]^{\circ}$. Assume that $b\in ...
0
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0answers
13 views

Extending orthogonal representation of $SU_2$ to $U_2$

Let $\phi: SU_2 \rightarrow SO_3(\mathbb{R})$ be the orthogonal representation of $SU_2$, obtained by letting $SU_2$ act on the three-dimensional vector space of trace-zero skew-hermitian matrices. ...
1
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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
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3answers
61 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
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2answers
44 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
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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
19 views

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

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
1
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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
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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 ...
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1answer
26 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
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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
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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
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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
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0answers
32 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
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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
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1answer
47 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
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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
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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
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1answer
58 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
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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
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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
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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
50 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
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0answers
26 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
37 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
58 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
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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 ...