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

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12 views

Is this AM-GM application correct?

I have got an inequality down to proving that: if $a,b,c$ are positive reals that satisfy $a+b+c=1$, then $$\frac{1}{1-\sqrt{a}}+\frac{1}{1-\sqrt{b}}+\frac{1}{1-\sqrt{c}}\ge \frac{9+3\sqrt{3}}{2}$$ ...
0
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0answers
23 views

Math Analogy with economics [on hold]

This may sound like a silly question, but I was reading something today and it said "Just as a hammer is not a building, but can be employed to build one, neither is mathematics economics, (clear ...
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1answer
31 views

Proof that $ax \equiv 1 \mod{n}$ has no solutions when $a$ and $n$ aren't co-prime?

Does this proof work? Is there a simpler one (precluding citing other theorems)? Suppose $ax \equiv 1 \bmod{n}$. Then $ax = kn + 1$. We have some $d = \gcd(a, n)$ such that $a = da'$, $n = dn'$, and ...
0
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1answer
70 views

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ [duplicate]

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ My Approach: I will be making use of $$\tag 1\quad{m+n\choose r} = {m\choose 0}{n \choose r} + {m\choose 1}{n\choose r- 1} + ...
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0answers
15 views

Every reflector between finitely complete categories is left exact…?

Let $\mathcal{A}\subseteq\mathcal{C}$ be a reflective subcategory, and $p\dashv i$ the related adjunction. Using that the counit $\varepsilon$ is an isomorphism (since the inclusion $i$ is fully ...
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1answer
50 views

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$ My Approach Let $x_k$ be one element in a set of $n$ elements. $n-1\choose r-1$ $=$ the number of unique groups of $r$ containing ...
2
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1answer
46 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
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0answers
80 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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0answers
26 views

Conditions any dense embedding from $(0,1]$ into $[0,1]$ must satisfy

This is a proof-verification request. Suppose that $m:(0,1]\to[0,1]$ is a dense embedding. That is, $m$ is continuous; $m$ is injective; the image $m\big((0,1]\big)$ is dense in $[0,1]$; $m$ has a ...
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1answer
36 views

Verify my proof: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $

Could someone verify my proof and my proof-writing? Proposition: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $ Proof: Suppose $ y $ is any positive natural ...
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1answer
22 views

Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
0
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0answers
164 views

Possible (trivial) error in Folland's “Real Analysis”?

Here's a link to the proof of Theorem 2.40. In the second displayed equation, shouldn't $<\epsilon$ be replaced by $\leq2\epsilon$? (And, the $3\epsilon$ right after, by $4\epsilon$?)
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1answer
20 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
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3answers
382 views

If D is an Integral Domain and has finite characteristic P, prove P is prime. Is my proof correct?

So the question is simply. If D is an integral domain and has finite characteristic prove that the characteristic of D is a prime number. This is my proof. Assume P is the characteristic of D. Let ...
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0answers
28 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from Introduction to Measure Theory by Terrence Tao. ...
4
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2answers
213 views

A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
2
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1answer
69 views

How to prove theorem using Euler's formula?

I'm having a great deal of trouble with this proof. "Prove $\cos θ + \cos 3θ + \cos 5θ + \cdots + \cos [(2n-1)θ] = \dfrac{\sin 2nθ}{2 \sin θ}$. Prove $\sin θ + \sin 3θ + \sin 5θ + \cdots + \sin ...
0
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2answers
54 views

Prove or disprove - If a divides b and b divides a does a=b

Prove or disprove: If a, b belong to the set of positive integers, and if a divides b and b divides a, then a=b. Does this hold if if a,b are not necessarily positive? Why or Why not? Here is what I ...
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2answers
57 views

how to prove $\sum_{i=1}^n i^k =\Theta(n^{k+1})$

we can say that if all $i$ s in the sum were equal to $n$ then the answer to the summation would be $n\cdot n^k$. So $n^{k+1}$ is the upper bound.so $\displaystyle\sum_{i=1}^n i^k=O(n^{k+1})$ For ...
0
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1answer
33 views

How can I show that a and b are odd in this contradiction proof?

Statement: suppose a,b belongs to Z (integers). If 4/(a^2+b^2) then a and b are not both odd. By proof of contradiction I assume that a and b are both odd. If a^2 and b^2 is odd then by definition a ...
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2answers
39 views

Is this contradiction proof correct?

Statement : suppose $a,b$ belongs to $\mathbb{Z}$ (integers). If $4/ (a^2 + b^2)$ then $a$ and $b$ are not both odd. Proof by contradiction: Assume that if $4/(a^2 + b^2)$ then a and b are both odd. ...
1
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1answer
25 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
6
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1answer
260 views

Is this proof good? Identifying extreme points of the unit ball in a function space

I want to prove: If $K$ is compact $T_2$ then the extreme points of the unit ball of $C(K)$ are precisely the functions $f\in C(K)$ such that $|f(k)|=1$ for all $k\in K$. Here is my proof: Can someone ...
0
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0answers
28 views

Can someone verify this proof? $\| x \|_2 \leq \| x \|_1$

Is this correct? $$\sum_{i \leq n} |x_i|^2 - \sum_{i,j \leq n} 2 |x_i||x_j| \leq \sum_{i \leq n} |x_i|^2 \implies \sum_{i \leq n} |x_i|^2 \leq \sum_{i \leq n} |x_i|^2 + \sum_{i,j \leq n} 2 |x_i||x_j| ...
0
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1answer
57 views

Correctness of proof that $\lim_{n\to \infty}\sqrt n*c^n=0$

My proof is as follows: Assume $|c|\lt 1$ and $c$ can be written as 1/1+d for d>0 The definition of the mentioned limit is: For all $\epsilon>0$ there exists a natural number N s.t. for all n ...
2
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1answer
23 views

Correctness of proof that an ordered field S that has the supremum property also has the infimum property

First question I have is how would you describe the relationship between an ordered field and an ordered set and continue the proof by treating the field as a set? I want to say that right in the ...
0
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1answer
24 views

Proof that the set of irrational numbers is dense in the reals

The hint I was given was to simply prove that y=xz is irrational given that x is nonzero, x is rational and z is irrational. Here's how I did it: Claim: y=xz is irrational Proof: Assume $x\neq0$, x ...
2
votes
1answer
208 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
1
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2answers
51 views

Is this calculus proof I came up with sound?

We want to prove there every bounded sequence has a converging subsequence. Let $[l,u]$ be the interval to which we know $a_n$ is bounded.Let $\{a_n\}$ be the sequence and $[l_i,u_i]$ where $i$ is a ...
2
votes
1answer
28 views

Algebra - proof verification involving permutation matrices

Theorem. Let $\textbf{P}$ be a permutation matrix corresponding to the permutation $\rho:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$. Then $\textbf{P}^t=\textbf{P}^{-1}.$ Proof. First note the following ...
0
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1answer
40 views

What does the notation $H=\{ a | a^2=e \}$ mean? [on hold]

Is it true that the notation $H=\{ a | a^2=e \}$ means $H=\{a,a^2=e\}$?
3
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2answers
80 views

Suppose $R \sim_\omega R'$. Then for every $k$-tuple $a$ in $E$ and every natural number $p$, there is a $k$-tuple $b$ in $E'$ such that $a \sim_p b$

Sorry to bother you guys again with a Poizat question, but I'm struggling a little bit with the material (as it must be obvious) and I want to check if I got the main idea correctly or if I'm totally ...
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2answers
54 views

Is this proof by induction for a sum of odd squares correct?

Statement: $1^2 + 3^2 + 5^2 + ... + (2n - 1)^2 = (n/3)*(2n-1)*(2n+1)$ Proof by induction -Base case: when $n = 1$ $1^2 = 1/3 * (2 * 1 -1) * (2 * 1 +1) = 1$ $1=1$ hence statement holds for $n = 1$ ...
3
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1answer
84 views

Question about proof of Browder, Minty Theorem

Could someone please assist with the following question: In the following SET OF NOTES, I am interested to know how the author obtains "By Lemma 1.11, the Galerkin equations (2.5 has a solution ...
2
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2answers
46 views

Proving differentiable function is continuous.

To prove that if function has a derivative at a then it is continuous at $a$, my teacher did: \begin{align} & \|f(a+h)-f(a)\|=\|f(a+h)-f(a)+f'(a)~h-f'(a)~h\| \\[8pt] \leq {} & ...
0
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0answers
29 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
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0answers
43 views

Basis of $\mathbf{Q}[x]$

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$-vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
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2answers
53 views

Proofread my work: Expressing generators of a cyclic group

The following question comes from Serge Lang's Undergraduate Algebra(pg. 26, 3rd edition). I just learnt the concept of groups and subgroups and I spent an hour or so on tackling part (b) of this ...
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1answer
35 views

Equivalence between two topological statements concerning the basis of a topology.

I need to show the following statement Let $\mathcal{B}\subset P(X)$ be a set of subsets of a set $X$, such that $\bigcup_{U\in \mathcal{B}}U =X$ then the following are equivalent $i)$ there ...
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1answer
18 views

Groups and U27 double check

This is just a quick question. The Group U$_{27}$=$(1,2,3,5,7,11,13,17,19,23)$ right? Or am I just very wrong here?
3
votes
2answers
56 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
5
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1answer
48 views

Show that there exists $\xi\in [a,b]: f(\xi)=\xi$.

Let $a,b\in\mathbb{R},~a<b$ and consider $f\colon[a,b]\to [a,b]$ continuous. Show that $f$ has a fixed point. i.e. that there exists a $\xi\in [a,b]$ with $f(\xi)=\xi$. My idea is to ...
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votes
2answers
32 views

$AB*\text{adjoint}(BA)=I$

$AB*\text{adj}(BA)=I$ Prove: $1$. $|AB|=1$ $2$. $AB=BA$ As for $2$. what I have menage is $AB*AB^{-1}=AB^{-1}*AB=AB*\text{adj}$(BA)=I$ \rightarrow BA=AB$ How do I solve $1$. and is $2$. is ...
0
votes
2answers
23 views

Is this the correct way to prove by induction?

Prove by induction that $$1 + 3 + 5 + 7 + ... + (2n + 1) = (n+ 1)^2 $$ //for every n ∈ $\mathbb N$. $$1+2+3+...+n=\frac{n(n+1)}2$$ Proof: $$3+5+7+\ldots+(2n+1)=$$ ...
0
votes
1answer
14 views

Prove that $F'(x) = \sum_{n=1}^\infty F_n'(x)$ almost everywhere.

Suppose $F_n$ is a sequence of increasing non-negative right continuous functions on $[0,1]$ such that $\sup_n F_n(1) < \infty$. Let $F = \sum_{n=1}^\infty F_n$ and suppose that $F(1) < \infty$. ...
0
votes
1answer
46 views

Proof of FTA from Hatcher

This is the proof of the fundamental theorem of algebra (FTA) given in Hatcher's Algebraic Topology textbook (I have underlined the relevant part): Could someone explain why $r$ needs to be ...
1
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0answers
22 views

Has the abc conjuncture been proved by Shinichi Mochizuki? [duplicate]

I'd like to know is his proof was reviewed, and what exactly happened.
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2answers
25 views

Contrapositive proof question, is this a valid way

Definition: $a\in \Bbb Z$ is a perfect square if there is a $b\in\Bbb Z$ and $a = b^2$ To prove: if $m$ and $n$ are perfect squares, then $mn$ is a perfect square. I know that this can most easily ...
1
vote
3answers
55 views

Proof by contrapositive: $x^3 + 1$ is even if and only if $x$ is uneven

$x^3 + 1$ with $x \in \mathbb{Z}$ is even iff $x$ is uneven. I want to prove this using a proof by contrapositive, so this is my work: Assume that $x$ is even, so $x = 2k$ with $k \in ...
2
votes
1answer
30 views

Statistics - Show that $\hat{\theta}$ hat is a biased estimator of $\theta$

I'm asked to solve this exercise, but I can't manage to find something satisfying. Any help/hint would be much appreciated. Let $Y_1, Y_2,\dots, Y_n$ denote a random variable sample of size n from a ...