For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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0answers
45 views

A combinatoric theorem about semigroups

I am struggling with the following theorem about semigroups, so I was hoping someone could give me a hand. The theorem states: "Let $S$ be an arbitrary semigroup such that for every $a\in{}S$ it ...
5
votes
2answers
481 views

Proof to sequences in real analysis

I need some verification for my proof in part a) and help to get me started on part b) a) Prove that the sequence $a_n = (2n+1)/(3n+5)$ converges to $2/3$ directly from the definition of convergence ...
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1answer
73 views

How do I prove: Let n ∈ N+. Let m ∈ N+. m<n. Prove that n⊥m ⇒ (n−m)⊥ m.

I don't even know how to start, any help would be greatly appreciated!
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1answer
450 views

Density of rationals and irrationals in real analysis

Can anyone help me solve this question? I've been working on it for two days already. Prove that for real numbers $x$ and $y$ with $x < y$, there is a rational and an irrational between $x$ and ...
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2answers
1k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
1
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3answers
34 views

Matrices iversion proof

Let $A$ be a square matrix such that $I-A$ is non-singular, prove that $A(I-A)^{-1}=(I-A)^{-1} A$ I can prove that $A(I-A)=(I-A)A$. But I dont know where to go next.
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2answers
88 views

Prove or disprove sequence is convergent

Let $a_n,b_n$ be two sequences of positive number, suppose that $\lim(a_n/b_n)=L$. Prove or disprove a) if $a_n$ is convergent, then so is $b_n$ b) if $b_n$ is convergent, then so is $a_n$ I start ...
4
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2answers
61 views

Proving convengent sequence theorem.

When $n$ approach to infinity prove that if $$ \lim(a_{n+1}-a_n))= 0,$$ then $a_n$ is convergent. I can prove the converse of this theorem is true but I can't prove this one. I know that since $$ ...
0
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1answer
79 views

Does integration over one complete cycle equals to 4 times integration over quarter-cyle?

From the article pendulum(mathmetics) from wikipedia. There is a demonstration that this equation: $$\dfrac{dt}{d\theta } = ...
0
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1answer
245 views

Intro to proofs in real analysis 3

Prove that for real numbers $x,y$ with $x< y$, there is a rational and an irrational between $x$ and $y$ in the following cases: a) when $x< 0< y$; b) when $x< y \le 0$. For a) this is ...
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3answers
59 views

non-existence limit proof

Find the limit and prove your answer is correct $$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}$$ By divide everything by $n^3$ I got $$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}=\frac10 $$ which is undefined. ...
0
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1answer
87 views

Proving basic limit laws without finding $\delta$s.

I'm brushing up on my calculus proofs, and I'm trying to show all the limit laws like $\lim_{x \to c} f(x) + \lim_{x \to c} g(x) = \lim_{x \to c} (f(x) + g(x))$, and similar for subtraction, ...
0
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2answers
93 views

Real Analysis Beginning Proof…

Alright, I've been assigned to work through a proof in my RA course and it just has me bogged down at this point. We're trying to show that If $b^2 > c$ then there exists a positive real number ...
0
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1answer
104 views

Prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2.

I am looking to prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2. So we nee that given any $\epsilon>0,\ \exists\delta>0$ so that $|f(x)-f(2)|<\epsilon\\$ whenever ...
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2answers
144 views

If $a$ is divisible by $4$, then there exist int $b$ and $c$ such that $a = b^2 - c^2$

I want to prove this: Prove that if $a \in \mathbb{Z}$ is divisible by $4$, then there exist $b$ and $c$ where $b,c \in \mathbb{Z}$ such that $a = b^2 - c^2$ I want to prove this directily: $4\mid ...
0
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2answers
7k views

Proof that when $x + y$ are irrational then $x$ and $y$ are irrational

I want to prove by contrapositive that: Proof that if $x + y$ are irrational then $x$ and $y$ are irrational. $x,y \in \mathbb{R}$ I did the following: Negation of the statement: $x + y$ are ...
1
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2answers
72 views

convergence proof

Let $k$ be a natural number. If sequence $b_n$ is obtained by deleting the first $k$ members of the sequence $a_n$, then $b_n$ is convergent if and only if $a_n$ is convergent. I know that if $a_n$ ...
2
votes
1answer
105 views

Is proof by contradiction always a sufficient proof technique?

Is proof by contradiction always a sufficient proof technique ? A proof by contradiction has the form: Let $P$ and $Q$ be statements. If $ P \rightarrow Q \land \lnot Q $ then you can conclude ...
0
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1answer
176 views

Rigorous proof of the properties of functions, limits, and relation to sequences.

This is all one big problem that builds on itself- meaning I cannot use results from later parts in parts before them. On the same token, I can totally use previous results in my proof of later ones. ...
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1answer
78 views

Induction Proof - Computation Theory [closed]

For my theory of computation class, we are supposed to do some review/practice problems to work off the rust and make sure we are ready for the course. Some of the problems are induction proofs. I did ...
0
votes
1answer
949 views

Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
0
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1answer
101 views

Constructing Proof Trees for Natural Deduction

I'm in the process of learning the process of writing so-called proof trees for $\textit{Natural Deduction}$. One question that I still grapple with is the actual process According to Van Dalen ...
0
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1answer
49 views

Prove that any completely regular semigroup $S$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$

Consider any completely regular semigroup $S$. I would like to prove, that any $a,b\in{S}$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$. So far, I was able to prove only the first ...
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0answers
228 views

Proof that $1729$ is the smallest taxicab number

For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and can ...
0
votes
1answer
72 views

Prove directly that $\forall x \in \mathbb{R^+} \exists y \in \mathbb{R^-}: y^2 = x$

I want to prove directly that the following statement is correct: $$\forall x \in \mathbb{R^+} \exists y \in \mathbb{R^-}: y^2 = x$$ By having a plain look I can say that it is true. However, what ...
1
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1answer
97 views

If $\Omega = \{1,2,3,\dots\}$ then $S_\Omega$ is an infinite group.

I don't think I quite get what the question is looking for. I wonder if anyone could point my attempt to the right track? Prove that if $\Omega = \{1,2,3,\dots\}$ then $S_\Omega$ is an infinite ...
1
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2answers
188 views

Real Analysis Proof Help

Prove that if a > 0 then there exists n as an element N such that 1/n < a < n. I know the answer involves the Archimedean Property, but I'm not sure how to write the proof.
5
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1answer
58 views

A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
4
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1answer
75 views

The 2 Ways to Prove Uniqueness - Interchangeable or Nonidentical?

An element belonging to some prescribed set $A$ and possessing a certain property $P$ is unique if it is the only element of $A$ having property $P$. Typically, to prove that only one element of ...
0
votes
1answer
364 views

Induction proof for the lengths of well-formed formulas (wffs)

Use induction to show that there are no wffs of length 2, 3, or 6, but that any other positive length is possible. The wffs in question are those associated with sentential/propositional logic. So, ...
1
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3answers
60 views

Uncertain how to proceed with combinatorics proof

The problem is as follows: let $n_1, n_2,..., n_t$ be positive integers. Prove that if $n_1+n_2+...+n_t-t+1$ objects are placed into $t$ boxes, then for some $i, i=1, 2, ..., t$, the $i$th box ...
5
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1answer
196 views

Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs

$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}$ $\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
0
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2answers
88 views

Prove $\left(\frac{1}{n}+\frac{(-1)^n}{n^2}\right)$ converges to $0$ as $n\to\infty$

Using the formal definition of convergence of a sequence, show that the sequence converges to 0 as n tends to infinity. So we want to show that for every $\epsilon>0$, there exists $N$ such that ...
1
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4answers
72 views

What does $\forall a \in A \exists b \in B(b \in C \rightarrow a \in C)$ mean?

What does $\forall a \in A \exists b \in B(b \in C \rightarrow a \in C)$ mean? I know it means that for all $a$ in $A$ there exists a $b$ in $B$ such that $b$ in $C$ implies $a$ in $C$, but what does ...
2
votes
1answer
261 views

Help to understand the proof of partial derivatives of homogeneous functions

I found this short proof that says the partial derivaties of homogenous functions of degree $k$ is homogeneous of degree $k-1$. Here is the proof in its entirety: I am lost at the very first step ...
2
votes
1answer
198 views

Contractible iff every map $f :X \to Y$, for arbitrary $Y$, is nullhomotopic. <Proof Verification>

I tend to write very inaccurate arguments even when my proof is correct. So I am wondering if someone would be willing to help me take a look at this proof? Thank you very much! (1) Show that a ...
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2answers
47 views

How to prove a line is above another line

Suppose I have the following line: $y=-4x + 80$, for $x \ge 0$ and $y \ge 0$ I want to show that if I vary the slope, $m$ like so: $-4\lt m \le -2$ Then the new line will be above the old line ...
0
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1answer
101 views

Is this a proof of why 0 isn't natural? [closed]

I have concluded here that the greatest lower bound of $A$ is negative. Is it a reasonable conclusion? I have also inferenced that this does not hold for the natural numbers. By the way, the original ...
3
votes
2answers
126 views

Prove that $2^n>2n$ for all integral values of n greater than 2 [duplicate]

Prove $2^n >2n$ for all integral values of n greater than 2. Let $p_n$ be the statement: $$2^n>2n\ \forall\ n\gt2$$ If the inequality is valid for $n=k$ where $k>2$: $$p_k: 2^k>2k$$ ...
2
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1answer
32 views

Is a direct product $\prod_{\alpha\in{}A}S_\alpha$ of semigroups $S_\alpha$ simple, if all semigroups $S_\alpha$ are simple?

I am currecntly trying to give an answer to the following problem. Consider a family of semigroups $(S_\alpha)_{\alpha\in{}A}$ and let every semigroup $S_\alpha$ be simple. Is it true or not, that ...
0
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2answers
53 views

Function $f(\textbf{x})=\|\textbf{x}\|^2\cdot \textbf{x}$ is of class $C^\infty$

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be given by the equation $f(\textbf{x})=\|\textbf{x}\|^2\cdot \textbf{x}$. Show that $f$ is of class $C^\infty$. I compute and find that ...
13
votes
2answers
197 views

Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
1
vote
1answer
228 views

Induction on a well-formed formula (wff)

Let α be a well-formed formula (wff); let c be the number of places at which binary connective symbols (∧, ∨, →, ↔) occur in α; let s be the number of places at which sentence symbols occur in α. (For ...
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votes
3answers
88 views

Prove that this inequality $5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$

Let $a,b,c>0$ and $a+b+c=1$ Prove that $$5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$$
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1answer
74 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
1
vote
1answer
83 views

Prove the rationals are arranged within the integers

I have a question about Terence Tao's lecture notes, Proposition 5: Let $x$ be a rational number. Then there exists a unique integer $n$ such that $n \leq x < n+1$. Here is my proof ...
7
votes
1answer
263 views

How to write well in analysis (calculus)?

This is kind of a subjective question, I know; often I find myself failing exams and homeworks because of the way i write down proofs. Either I don't know how to start, or somehow the main point of ...
2
votes
2answers
439 views

Infinum & Supremum: An Analysis on Relatedness

$\require{color}$ Question: I need some help in proving that $\color{green}{\text{if $k\geq 0$, then $\sup (kS) = k\sup(S)$ and $\inf (kS) = k\inf (S)$}}$, and also that $\color{red}{\text{if ...
1
vote
0answers
74 views

Stylish Academic Writing [duplicate]

I don't know that this question belongs here, but I'd like to know of any references out there anyone here might recommend for writers of mathematical ideas, be it a book, an article, a dissertation, ...
3
votes
2answers
100 views

Prove that $n + 2$ is odd where $n = 2k+1$ for some integer $k$

I am attempting to learn about mathematical proofs on my own and this is where I've started. I think I can prove this by induction. Something like: $n = 2k+1$ is odd by definition $n = 2k+1 + 2$ ...