For questions about the formulation of a proof, not about the mathematics behind it.

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0answers
21 views

Is every Complex Square Matrix similar to its transpose? [duplicate]

I am aware that every complex square matrix is similar to its transpose but I am having a hard time proving this. Should I try to use the previously asked question listed at $A matrix is similar to ...
1
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1answer
41 views

Can I mix direct proof with inductive proof?

Let's say I want to prove with induction that $3|n$ implies $3|n^2$ Let $n = 3k$. The statement is true for $k=1$ since $3|3$ and $3|9$ We assume the statement is true for $k=z$ so $3|3z$ ...
1
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1answer
33 views

An approximation for the Lambert W-function

Proposition Let $f(x) = k^{x}x$, where the values of both $f(x)$ and $k$ are known. Let $x_{0} = f(x)$, and: $$x_{n + 1} = \frac{1}{2}\log_{k}{\left(\frac{k^{x_{n}}x_{0}}{x_{n}}\right)}$$ ...
1
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2answers
63 views

Prove that the set [a,b] is not well ordered. where a,b are real numbers.

My Proof: Assume towards contradiction that [a,b] is well ordered. (a,b) is a subset of [a,b]. Thus (a,b) has a least element. Let's call this element m. We know that: $m>a \\ m-a>0 \\ ...
0
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1answer
28 views

Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable.

Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable. I am having trouble figuring out where to start. I know that if I show that A + αIn has n distinct real ...
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3answers
44 views

Prove or disprove that $(a_n)_{n=1}^{\infty}$ is Cauchy $\iff$ $\displaystyle\inf_{n \ge 1}{\sup_{k,l \ge n}{|a_k - a_l|}} = 0$

(How to state that a sequence is Cauchy in terms of $\limsup$ and $\liminf$? For the above post by me, I have this new claim and its unfinished proof, but I am not sure should I edit that old question ...
3
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1answer
53 views

Writing proofs with modular arithmetic

I am enrolled in Discrete Mathematics 2 and I am having trouble understand a lot of the material. For the particular problems I need help with I need to: Prove each of the given statements, assuming ...
0
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1answer
23 views

Question about a proof of $ g(A\cap B)\subset g(A)\cap g(B) $

I was trying to prove: $$ g(A\cap B)\subset g(A)\cap g(B) $$ which has been answered lots of time on here but I had a question about a part of my attempted proof (which is hopefully correct, I'm ...
2
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2answers
76 views

Demonstrating the image of the inverse image of a subset

I need to demonstrate the following: Let $E, F$ be sets, $Y \subset F$ and $f : E\longrightarrow F$. Prove that $f(f^{-1}(Y)) = Y \cap f(E)$ I tried do prove that using double inclusion with ...
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1answer
41 views

How to prove algebraic theorem of the solution set of the equation ax+b=c using field axioms

Prove the following Theorem: If a, b, c are numbers, the solution set of the equation ax + b = c consists of either (a) a single number, (b) the empty set, or (c) the entire real line. Hint: If you ...
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1answer
54 views

How to improve my proof on monotone sequences converging if bounded

My professor gave me 3/5 points for this proof and just wanted to know how I could make it better. Prove a monotone sequence converges if and only if it is bounded. If {$x_n$} is monotone then it ...
0
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1answer
32 views

Proof by Contradiction

Prove by Contradiction: Let $a,b,k$ be an element of $\Bbb Z$. If $a|k$ or $b|k$ then $(ab)|k$. How should I proceed? I have $a=\frac{k+s_1}{l}$, $b=\frac{k}{r}$ and $k=abp+s_2$ where ...
0
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1answer
51 views

Negate the following definition of the concept of limit [closed]

The following is the definition which I am trying to negate but I cannot wrap my head around it
4
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1answer
94 views

Proof that $(t_1, \dots, t_r) \mapsto \sum^{r}_{i=1} | t_i - \alpha_i|^p$ is continuous - Problem with Inequality

Bounty Edit: I already edited the question after some important comments. The questions I have are highlighted below the supposed proof. Any feedback or answer is most welcome. Thus, I just found a ...
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2answers
23 views

Cardinality proof verification

Problem: Let $C \subset (0,1)$ be the set of all numbers whose unique decimal representation contains the number seven. Show that the number of elements in $C$ must be the same as the number of ...
4
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0answers
139 views

I have a free summer before university. What should I learn? [closed]

Note: This is a soft question. It may be a bit early to be thinking about this, but I figured I'd ask now and see what responses I get. I'm currently a high school senior, and I quite like pure ...
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4answers
46 views

Prove $\sum\limits_{i=1}^n \frac{1}{i(i+1)} = \frac{n}{n+1}$ by induction [closed]

Using induction, prove that $$\sum\limits_{i=1}^n \frac{1}{i(i+1)} = \frac{n}{n+1}$$ Any help would be appreciated.
1
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1answer
105 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
2
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2answers
33 views

$\bigcup\limits_{i=1}^n A_i$ has finite diameter for each finite $A_i$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be $\operatorname{diam}(A)= \sup\{d(x,y):x,y\in A\}$. Suppose $A_1, \dots, A_n$ is a finite collection of subsets of ...
0
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2answers
27 views

Check my proof of showing that diam$(A)=$ diam$(\bar{A})$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be diam$(A)=$ sup$\{d(x,y):x,y\in A\}$. Show that for any set $A\subset X$, diam$(A)=$ diam$(\bar{A})$ where $\bar{A}$ ...
2
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3answers
111 views

Prove that there exist four distinct real numbers a, b, c, d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are irrational

So i am doing this example and i found a way to prove there exists that but it seems ugly. Can you guys help me find a better solution? My proof: a*b is irrational when one of them is irrational and ...
1
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1answer
32 views

Proof real number $\sqrt{3}$ is an irrational number [duplicate]

Given: Let $n$ be an integer. Then $n^2=3a$ for some integer $a$ if and only if $n=3b$ for some integer $b$. Proof real number $\sqrt{3}$ is an irrational number. Here is I have so far: Assume ...
1
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1answer
81 views

Interpretation of 2 proofs involving limits at infinity and mathematical induction

I have 2 exercises that I think are related to each other. I think they should be proved by mathematical induction. They are: prove that: limit of n which approaches infinity $(2^n / n!) = 0$ prove ...
1
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1answer
69 views

Show that for every set of 18 integers there will be two that are divisible by 17 [closed]

I understand the pigeonhole principle is needed here and I see the solution in the back of the book, but the explanation is week. If anyone could explain step-by-step that would be awesome!
0
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1answer
156 views

proving a hard inequality [closed]

can someone help me to prove this inequality : $\left| \sum _{ k=0 }^{ 2n }{ \frac { k }{ k+{ n }^{ 2 } } } -\sum _{ k=0 }^{ 2n }{ \frac { k }{ { n }^{ 2 } } } \right| \le \frac { 4 }{ { n }^{ 2 } ...
1
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1answer
53 views

Proof by cases: If n is a positive integer and there is a perfect square in $\{k \in \mathbb{Z} | n \leq k \leq 2n\}$

If n is a positive integer and there is a perfect square in $\{k \in \mathbb{Z} | n \leq k \leq 2n\}$, then there is a perfect square in $\{k \in \mathbb{Z} | n + 1 \leq k \leq 2n + 2\}$. There seems ...
0
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1answer
36 views

Do $P(X\cup F)=P(E) \cup P(F)$?

Given $E$ and $F$ two sets And $P$ is part of set Do $P(E \cup F)=P(E) \cup P(F)$ ? $X\in P(E \cup F)\ \Longrightarrow\ X\in P(E)$ or $X\in P(F)$ Im stuck here.
4
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3answers
362 views

How to write mathematical induction?

Reading the literature about mathematical induction, I have learnt that there are between 4 and 3 steps in reasoning and writing the proof. I say between 3 and 4 because actually I see that texts and ...
2
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2answers
36 views

Prove root of unity and order

I have this math problem: i) Suppose that $a$ and $b$ are roots of unity. Suppose that $o(a)=5$ and $o(b)=7$. Prove that $o(ab)=35$. ii) Give an example such that $a$ and $b$ are roots ...
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3answers
31 views

Show root of unity and order

I have this math problem: Set $$z=\frac{1}{2}-\frac{\sqrt{3}}{2} i$$ Show that $z$ is a root of unity, find its order, and express $z^{100}$ in the form $a+bi$. I'm not 100% sure how to do ...
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1answer
29 views

TOPOLOGY by Munkres Lemma $58.4$

I can understand the steps but I can't understand how . The problem is their approach is not clear to me, why he did it that way . When $h$ and $k$ are given to be homotopic ...
0
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2answers
28 views

Demonstration with complex numbers by using conjugates,

Let $\text{conj}$ be the complex conjugate. (It makes the following fraction look nicer than $\bar z$.) So I must demonstrate the following : $$\text{conj} \left( \frac{1}{z} \right) ...
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0answers
46 views

finding counterexamples to proofs.

let f: A → B and let W, X ⊆ A. Prove that if W ⊆ X, then g(W) ⊆ f(X) I don't see how this can work, as I think I've found a counter example. Yet the instructions ask for a proof. Let A = {0,1,2} ...
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1answer
30 views

Show root of unity summation

I have this math problem Let $w$ be a root of unity with $o(w)=n$, with $n > 1$. Show that $$1 + w + w^2 + \cdots + w^{n-1} = 0$$ I'm not entirely sure how to start this problem. Would I ...
0
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1answer
31 views

proof check product equals zero entails a multiplicand is zero

Currently trying to learn/teach myself proofs, however I could use some feedback on this proof. just want to know if its correct. I am sure there is a much quicker way to prove this. My main concern ...
0
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3answers
32 views

Prove complex number relationship with triangular inequality

I have this math question that I'm kind of stuck on. Suppose that $a$ and $b$ are complex numbers. Use the result of the triangular inequality to prove that $$ |a| - |b| \le |a - b| ...
0
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2answers
35 views

Show complex numbers equal each other

I have this math question: Let $z$ be a complex number and $a$ a complex number with $|a| = 1$. Show that: $$|1 - a\overline{z}| = |a - z|$$ So far I have this: $$\mid ...
0
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1answer
31 views

Proof by Contradiction/Contrapositive

Hi there I'm just starting to learn about Proofs and I'm not sure how to prove the example below. Suppose A\B ⊆ C∩D. For every x ∈ A, if x ∈/ D then x ∈ B. Any advice would be wonderful. Thanks!
3
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1answer
26 views

Proving angles in the same corner equal

Suppose we have two line segments, AB and CD, which cross at point X. Now suppose there is an arbitrary point Y somewhere on the segment AX (that is, points A, Y and X are collinear). What is the ...
1
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2answers
54 views

Subgroups of $S_4$ (proof)

How do I prove that $S_4$ has a subgroup of order $d$ for every factor $d$ of $24$? Hence, $S_4$ must have eight subgroups with orders $1,2,3,4,6,8,12,24$. It's clear that $\{e\}$ is a subgroup of ...
3
votes
2answers
51 views

At what point is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous?

Define $f(x)=\begin{cases}x^2\,\,\,\,\text{if $x\leq 0$}\\ x+1\,\,\,\text{if $x>0$}\end{cases}$At what point is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous? Justify the answer. I ...
4
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2answers
107 views

There is a free group $F_2$ in $SO(3)$

I'd like an exposition of the proof that there is a subgroup of $SO(3)$ isomorphic to $F_2$, the free group on two elements. This is a key step in the proof of the Banach-Tarski paradox. The usual ...
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1answer
31 views

Proof: Sigma Algebra

Let $f:X \to Y$ be a function and let S be a sigma algebra on Y, that is, S is: i) Closed under countable unions ii) Closed under complements iii) $\emptyset \in S$ Show that $R = ${$f^{-1}(B) : ...
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votes
2answers
95 views

Show that: $\sqrt{abcd}\le\left(\frac{a+b+c+d}{4}\right)^2$

Assuming that $a,b,c,d$ are not negative, How can one show that: $$\sqrt{abcd}\le \left(\frac{a+b+c+d}{4}\right)^2$$ I tried this: By $\text{AM}\ge \text{GM}$ we have ...
0
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1answer
50 views

Is this a valid Answer?

First off, I am quite open to changing the name of the question if anyone has suggestions, so that it might be more accessible and helpful to future mathonaughts. I need to describe partitions for ...
1
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3answers
54 views

Epsilon-n proof confusion

So, I'm told I did something wrong on this epsilon-n proof but I really don't know what. I'm told to prove that $$\lim_{n\to\infty} \frac{5n+1}{n+3} = 5 $$ I begin by setting $\epsilon > 0$ and ...
0
votes
3answers
42 views

$x^2+y^2+2axy=0 \Rightarrow x=0$ and $y=0$

Show that for all real numbers $x$ and $y$, for all $-1<a<1$ $$x^2+y^2+2axy=0 \Rightarrow x=0 \text{and} y=0$$ I see that $x^2+y^2+2axy=(ax+y)^2+x^2-(ax)^2$. I'm stuck here.
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0answers
43 views

Baby Rudin exercise 4-19, is there anything wrong with this solution?

Associate to each sequence $a=\{\alpha_n\}$, in which $\alpha_n$ is $0$ or $2$, the real number $$x(a)=\sum_{i=1}^{\infty}\frac{\alpha_n}{3^n}$$ Prove that the set of all $x(a)$ is precisely ...
0
votes
0answers
24 views

proving the limits of the product of two functions

Let $x_0$ be a real number and suppose that $f(x)$ and $g(x)$ are real valued functions defined for all $x>x_0$. If $\lim \limits_{x \to x_0+}f(x)=\infty$ and $\lim \limits_{x \to ...
0
votes
2answers
51 views

Proof of the Reverse Triangle Inequality

Here there is my proof (quite short and easy) of a rather straightforward result. The text of this question comes from a previous question of mine, where I ended up working on a wrong inequality. Here ...