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

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27 views

Prove $\sqrt s$ exists by the Intermediate Value Thorem?

I'm pretty sure the process is right, but I'm not sure if I've validly presented the proof. Here is what I wrote: Suppose that $s > 0$ and consider the function $f(x) = x^2$ on the interval ...
1
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1answer
23 views

Surjectivity and the non-existence of maps.

This question comes from Jacobson's Basic Algebra. It asks: Show that $S \overset{\alpha}{\to} T$ is surjective iff there exist no maps $\beta_1,\beta_2$ of $T$ into a set $U$ such that $\beta_1 ...
0
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1answer
29 views

Increasing real valued function whose image set is connected

Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true: a) $f$ is discontinuous exactly at one point. b) $f$ is ...
2
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2answers
44 views

Show that $\hat{\theta}$ is an unbiased estimator of $\theta$

Let $f(x, \theta) = \frac{1}{\theta} x^{\frac{1-\theta}{\theta}}$, where $0 < x < 1$ and $\theta > 0$. Let $X_1, \dots, X_n$ be iid with density $f$. Taking the log likelihood, I found $$ ...
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0answers
40 views

Properties of ODE without solving

I stumbled over a nice exercise in a Textbook about ODEs, witch i couldn't figure out yet. Here's the Problem $ u''=-u-u^{3}-ku'$ $,k >0$ show that for every k there exists a nontrival solution, ...
1
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1answer
49 views

Prove that there doesn't exist prime numbers $a, b, c$ s.t. $a^2=b^2+c^3$

I first showed that if $a,b,c \neq$ 2, then they are odd and therefore are never equal. Then I consider the cases where $a=2$, $b=2$ and $c=2$. It seems to be unnecessarily long so is there a more ...
0
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1answer
24 views

Please find errors in my reasoning about field axioms

We can define a field F with the following properties: Binary operations + (addition) and ⋅ (multiplication) Commutativity Associativity Identities Inverses Distributivity Now, the additive ...
2
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0answers
46 views

How to analytically continue this function?

I was wondering if it would be possible to get an analytically continuation of the following function: $$ J(x) = \sum_{r=1}^\infty \ln(r)x^r $$ My attempt Consider the following: (1) $$ J'(x) = ...
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2answers
41 views

If a holomorphic map $f$ has constant real part on some ball $B \subseteq \Omega$, then $f$ is constant on $\Omega$.

I would like to know if my reasoning is correct. I tried to prove the following : Let $\Omega \subseteq \Bbb C$ a connected open set, $f : \Omega \to \mathbb C$ a holomorphic function, such that ...
0
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1answer
28 views

Proof for length of graph

G is a simple graph that consists of a vertex set V(G) = {v1, v2, ..., vn} and an edge set E(G) = {e1, e2, ..., em} where each edge is an ordered pair of vertices. The edge {u,v} is denoted uv. A ...
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0answers
30 views

If $z=2(\cos \theta + i \sin \theta)$, use the triangle inequality to find an upper estimate for $|e^{z^2} + 4\sin(z)|$

If $z=2(\cos \theta + i \sin \theta)$ ($0 \leq \theta \leq 2\pi$), use the triangle inequality to find an upper estimate for $|e^{z^2} + 4\sin(z)|$. Okay so if I write $z$ in the form $x + iy$ I ...
0
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1answer
190 views

Prove 1 is not the largest integer? [duplicate]

This proof looks extremely flawed, but I'm new to proofs so I'm not completely sure what is allowed and what isn't. Here it is: Let $n$ be the largest positive integer. Then $n$ must be $\geq 1$. ...
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2answers
58 views

Prove $\frac{2ab}{a+b}\leq\sqrt {ab}$

$a$ and $b$ are both positive real numbers. I'm supposed to work backwards (i.e. start with what I'm trying to prove and change it until something is absolutely true, then start from what is ...
6
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1answer
43 views

Prove that $\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$

Prove that: $$\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$$ I am self-learning these stuff, and I would like to check whether I did things right. Here's ...
0
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2answers
46 views

If true, prove that 2 real numbers satisfy $a<b$ iff $a<b+ \epsilon$ $\forall \epsilon >0$

I get that this is a biconditional statement, it holds in the forward direction since $\forall a<b$, $a<b+ \epsilon$ $\forall \epsilon>0$ In the reverse direction I get confused: It seems ...
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0answers
48 views

Limits of two fixed points of $E_\mu(x) = \mu e^x$

Please let me know if this proof is OK. Problem statement: Given that $E_\mu(x) = \mu e^x$, where $0 < \mu < 1/e$, show that if $q_\mu < p_\mu$ are fixed points, where $q_\mu$ is attractive ...
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0answers
50 views

Need to verify Real Analysis Proof

I wish to verify a proof; solutions to this exercise are not available. Lemma: If $S \subset \mathbb{Z}$ is bounded from above, it has a maximum element. Proof: If $S$ is bounded from above, it ...
1
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2answers
47 views

Use the Intermediate Value Theorem to prove that $\sqrt s$ exists?

I need to prove with the Intermediate Value Theorem that $\sqrt s$ exists, where $s > 0$. My textbook states this definition of the Intermediate Value Theorem: Suppose that $f$ is continuous on ...
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0answers
17 views

Verification on classification of singularities

In an exercise, I'm asked to classify the singularities of these functions: $\qquad i) f(z)=\frac{1}{(z-1)^2} \qquad ii)f(z)=\frac{1-\cos z}{z^2} \qquad iii) f(z)=\frac{z^2-1}{z-1}$ I don't know why ...
1
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2answers
53 views

$\cot^{-1}x = \tan^{-1}\dfrac{1}{x}$ Why are my methods of proving this true are wrong?

$\cot^{-1}x = \tan^{-1}\dfrac{1}{x}$, $\forall$ $x\in R -$ {$0$} Say I need to check if its true or not. METHOD 1 Putting $\cot^{-1}x = \pi/2- \tan^{-1}x$ $$\Rightarrow \pi/2- \tan^{-1}x = ...
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0answers
30 views

Exercice about the group of unit $(\mathbb Z/21\mathbb Z)^\times $.

Let $(\mathbb Z/21\mathbb Z)^\times$ the group of units of $\mathbb Z/21\mathbb Z$. 1) How many element in $(\mathbb Z/21\mathbb Z)^\times$ ? 2) Is it isomorphic to an abelian group ? 3) Is it ...
0
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1answer
93 views

Proving by induction that a balanced strings of parentheses has equally many opening and closing parentheses

In computer science, a string is a finite sequence of characters. For strings $A$ and $B$, we express $AB$ as $A$ followed by $B$. A balanced string of parentheses is a string of open and closed ...
1
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1answer
52 views

Is my choice of $\delta$ under the given bounds for M correct in this $\epsilon-\delta$ proof? Are the choices of < or = correct?

Prove that $$\lim_{x \to 9} \sqrt{x} = 3$$ Let $\epsilon > 0$. $\color{red}{\text{Choose} \ \delta = \min\{\epsilon(\sqrt{9-M} + 3), M\} \ \text{where} \ M \in (0,9)}$. Case 1: ...
0
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1answer
48 views

Prove that every topology in X is a basis for itself.

Here's my proof: Let $\mathcal{T}$ be a topology and $x \in X$. Since $\mathcal{T}$ is a topology on $X$, then $x \in X \in \mathcal {T} \Rightarrow X \subset \bigcup B_x \Rightarrow X = \bigcup ...
3
votes
2answers
58 views

Prove $|x|^2$ = $x^2$

My first attempt at this proof divided into 2 cases, one where $x^2$ is greater than or equal to 0, and another where $x^2$ is less than 0. For the first case, I said that the definition of absolute ...
4
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1answer
19 views

Number of quadrilaterals in a heptagon: is my reasoning correct?

I found this question on a GRE prep site: If you join all the vertices of a heptagon, how many quadrilaterals will you get? There is a bunch of multiple choice answers but to me none of them ...
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2answers
40 views

Finding error in an incorrect proof

Statement: If $a$, $b$ and $b'$ are integers and $a>b>b'>0$, then the remainder when $a$ is divided by $b$ is less than the remainder when a is divided by $b'$. Proof: Assume $a,b, b'$ are ...
5
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1answer
45 views

What is the probability that the upturned faces of three fair dice are all of different numbers?

Three fair dice are rolled ($6$ sides). What is the probability that the upturned faces of the three dice are all of different numbers? I got that the number of possible outcomes total is $6^3$ ...
0
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1answer
61 views

How to express the function $\mathbb{N} \to \mathbb{N}\times \mathbb{N}$ as a mathematical statement?

I am not that good at creating proofs, but I am decent at coding and was able to come up with this simple "program" that makes up the function I desire: ...
3
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4answers
103 views

The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$

Regarding real numbers, the following appears to be true, or at least true with some modifications. Could you help me for the proof? $$\int_0^af(x)dx+\int_{f(0)}^{f(a)}f^{-1}(x)dx=af(a)$$
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0answers
53 views

Question about the induced Hurewicz isomorphism

In my notes it's claimed that the group homomorphism $$\Phi: \pi_{1}(X,x_{0}) \to H_{1}(X), \space \{f\} \mapsto[f]$$ clearly induces a group homomorphism $\Phi_{*}: \pi_{1}(X,x_{0})^{ab} \to ...
0
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1answer
20 views

Sum of measures of non-disjoint sets

I am solving problem 3.3.10 from Royden 3rd Edition Real Analysis. The solution manual has a simple, one-line proof for this simple problem, but I had come up with a longer version that involves the ...
0
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2answers
29 views

Proof of Direct Sum of Subspaces

Given a space $V=\mathbb{R}^3$ I have proved that the plane $M= \{ (x,y,z) \in \mathbb{R}^3 \mid x+y+z=0\}$ and the line $N:\{ (x,y,z)\in \mathbb{R}^3 \mid x=-\frac{3}{4}y=3z\}$ are both subspaces of ...
0
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1answer
34 views

How do you define such map $(C^B \times B^A) \to C^A$?

Suppose that $\mathbf{C}$ be cartesian closed and $B$ is an object of it. We define two functors $\mathbf{C} \times \mathbf{C} \to \mathbf{C}$ by $$ C^B \times B^A \qquad\text{and}\qquad ...
12
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2answers
132 views

Prove $\int_{0}^{x}f+\int_{0}^{f(x)}f^{-1}=xf(x)\qquad\text{for all $x\geq0$}$ [duplicate]

Suppose that the function $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous and strictly increasing and that $f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable. Moreover, assume $f(0)=0$. ...
0
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0answers
31 views

Real Analysis, Folland problem 3.2.14 The Lebesgue - Radon-Nikodym Theorem

Relevant background information: We say that two signed measures $\mu$ and $\nu$ on $(X,M)$ are mutually singular if there exists $E,F\in M$ such that $E\cap F = \emptyset$, $E\cup F = X$, $E$ is ...
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1answer
31 views

Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...
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0answers
23 views

False Counterexample for “for all sets A, B, and C, A ∩ (B - C) = (A ∩ B) - (A ∩ C)”

I've put together a proof on this, (which I would appreciate being verified), but I also want to know what a false counterexample might be for this? I'm new to discrete mathematics, and I'm honestly ...
0
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0answers
31 views

Prove $\inf(-x) = -\sup(x)$

Suppose S is bounded above, Prove $\inf(-x) = -\sup(x)$ I've reckon there are similar proofs here with negative signs in different ways, but I want to get some proof verification as well as some ...
1
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2answers
41 views

Simple Inequality of Complex Numbers, $\left| \frac{a-b}{1-\overline{a}b} \right| <1$

Exercise from Ahlfor's Complex: Given $a,b \in \mathbb C$, with $|a| <1$, $|b|<1$, prove: $$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$ My argument: Lemma: If $\alpha, \beta \in ...
4
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1answer
74 views

From marginal distribution to joint distribution

Consider two sequences of real-valued random variables, $\{X_n\}_n$ and $\{T_n\}_n$. Let $\rightarrow_d$ denote convergence in distribution. Assume (1) $X_n\rightarrow_d L$ as $n\rightarrow \infty$, ...
1
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3answers
40 views

Using a Direct Proof to show that two integers of same parity have an even sum?

I seem to be having a lot of difficulty with proofs and wondered if someone can walk me through this. The question out of my textbook states: Use a direct proof to show that if two integers have ...
1
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1answer
48 views

Prove that $\lim\limits_{n \to \infty} f(x_n) = f(L)$

If $f$ is continuous on $[a,b]$, if $a≤x_n≤b$ for each $n$, and if $\lim x_n = L$, prove that $\lim f(x_n)=f(L)$ To start with, I have already proven in a previous assignment that if if $a≤x_n≤b$ for ...
1
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3answers
55 views

If $E\subset \mathbb{R}$ is bounded then $x=\sup E$ is in $\overline{E}$

Please let me know if you think my proof is rigorous enough. Notation: $\overline{E}$ - the closure of E; $\partial E$ - the boundary of $E$; $E^\circ$ - the interior of $E$. If $E$ is closed then ...
0
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0answers
64 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z}^{*}_{p},\cdot)$ with the ...
0
votes
1answer
31 views

Proof of $\sup \epsilon x = \epsilon \sup x$

Suppose S is a non empty set of real numbers, and suppose S is bounded above, and that $\epsilon > 0$, Prove $\sup \epsilon x$ = $\epsilon\sup x$ My take so far: $sup S = B$, then $B$ is an ...
0
votes
2answers
23 views

Prove that a denumerable set can be partitioned into two denumerable subsets

I was wondering if this "proof" is sufficient in demonstrating a that a denumerable set $A$ can be partitioned into two denumerable subsets $A_1$ and $A_2$. Let $A$ be a denumerable set and define $A ...
0
votes
1answer
38 views

If a set is proper and non-empty then its boundary is non-empty

Please let me know if this proof looks good. For the sake of contradiction, suppose that $E\ne \emptyset$ and $E\ne \mathbb{R}^n$, and that the boundary of $E$, $\partial E$, is empty. Since the ...
2
votes
1answer
25 views

Set $C$ contains set $B$ which is the largest subset of $C$ relatively open in $E$

Please let me know if you think this proof is OK. Given a set $E\subset \mathbb{R}^n$ and $C\subset E$, prove that $C$ contains a set $B$ which is the largest subset of $C$ relatively open in $E$. ...
2
votes
1answer
44 views

The union of two disjoint closed sets can't be an interval

Let $F,G$ be subsets of $\mathbb{R}$, both closed and disjoint. Suppose their union is a closed interval. Then at least one of them is empty. Well, I've got some doubts with my proofs(I wrote two of ...