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

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Relatively Prime Relationship Equation Proof

I have this math question that I am stuck on. This is the question: Suppose that $a$ and $b$ are positive integers, with $d = \gcd(a, b)$. Suppose that there exists integers $r$ and $s$ so ...
2
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2answers
79 views

Proving Theorems

I've been struggling with the concept of proofs ever since I completed my introductory logic course "Axiomatic Systems". In that course it seemed to be easy. We were pretty much just using various ...
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0answers
34 views

Proof that the collection of all finite subsets of $\mathbb{N}$ is countable.

Just looking for some feedback on my proof. Am I missing anything? Thanks! Suppose $A$ is the collection of all finite sets in $\mathbb{N}$. Trivially we have $\emptyset = 1$. Define function f on A ...
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1answer
20 views

Show simple continued fraction with Euclid's Algorithm

I have this math problem, I have to show a simple continued fraction from equations. Here's the question. Use these equations: $$397 = 204(1) + 193$$$$204 = 193(1) + 11$$$$193 = 11(17) + 6$$$$11 ...
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1answer
15 views

Prove with Euclid's Algortihm

I have this problem and I'm not 100% how to complete it. Here's the question: Let $m$ and $k$ be positive integers with $m > 1, k > 1$. Show that $\gcd(m, mk - 1)=1$. (Hint: use ...
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0answers
25 views

Prove that $AB\mid CD$

I have this math question that I'm kind of confused on. This is the question: Let $A, B, C$ and $D$ be integers with $A \mid C$ and $B \mid D$ show that $$ AB \mid CD. $$ I'm not 100% sure ...
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3answers
92 views

Prove using induction

I have this math problem I'm kind of stuck on. Here's the question: Define a sequences of real number with the definitions $$\begin{align*} x_1 & = 3 \\ x_n &= \sqrt{2 x_{n-1}+1} ...
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3answers
60 views

$\int_0^c f(x)dx=0$ for each $c\in[0,1]$ then $f=0$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous and $$\int_0^c f(x)dx =0$$ then prove that $f\equiv 0$.Do not assume that $f\ge 0$. How do I do it $?$ . I have this slightly geometric ...
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1answer
14 views

proofs involving the triangle inequality.

$$\forall a,b \in R (|a+b|=|a|+|b| \iff ab \ge 0)$$ I'm really stuck on where to even start with this. I'm assuming it has something do to with the triangle inequality, but don't know how to apply ...
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2answers
44 views

Strange induction proof

I'm trying to solve an induction proof exercise but this time I can't even understand how to proceed. I must prove that for every given $n\in \mathbb{N}$ with $n\geq2$ there exist $a,a_1,a_2,...,a_n$ ...
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1answer
42 views

What continuity of the derivative has to do with the proof of $\int_a^b f +\int_{f(a)}^{f(b)} f^{-1} =bf(b)-af(a)$

Question: Let $f:[a,b]\rightarrow \mathbb R$ be continuously differentiable and $f'\gt 0$. Prove that $$\int_a^bf +\int_{f(a)}^{f(b)} f^{-1} = bf(b)-af(a) $$ My problem with this : $f'\gt ...
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4answers
51 views

counterexample to a proof.

Prove the following statement; $$\forall a,b \in R (\forall \epsilon > 0 (a \le b + \epsilon) \rightarrow a \le b)$$ I can't see how this is true This means that I can pick a number for all ...
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2answers
30 views

Show that all elements of one sequence are less than all elements of another sequence.

Let $\{a_n\}_1^\infty$ and $\{b_n\}_1^\infty$ be two sequences in $\mathbb{R}$ such that $\forall n \in \mathbb{N}$, it is true that $a_n \leq b_n, a_n \leq a_{n+1}, \text{and} b_{n+1} \leq b_n$. We ...
3
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2answers
83 views

Show that sup$AB$=(sup$A$)(sup$B$)

Where $AB$ is the product of the sets and $A,B \in \mathbb{R^+}$. Since $A,B$ are bounded above sup $A$ and sup $B$ exist. Let $\alpha = $ sup $A$ and $\beta = $ sup $B$. This implies $\forall a \in ...
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2answers
63 views

How to prove prime power factorization is square free

The question is as follows: "Show every positive integer is the product of a square (possibly 1) and a square free integer" We begin by writing a positive integer n in its refined prime power ...
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1answer
40 views

Strengthening an inequality

I'm reading a book and there's an example problem that goes like this: Prove that $$ \left(\frac{1}{2}\right) \left(\frac{3}{4}\right) ... \left(\frac{2n-1}{2n}\right) \le \left(\frac{1}{\sqrt ...
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1answer
64 views

Prove that the following products of prime numbers equals to infinity

I can't understand how to show that the product $\prod_{p \equiv 1(mod 3)}\frac{p}{p-1}=\infty$ and $\prod_{p \equiv 2(mod 3)}\frac{p}{p-1}=\infty$ I was shown a similar proof without the ...
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0answers
61 views

show that there are infinitely many primes congruent to 1 or 4 modulo 5…

Given the following Dirichlet character: $\epsilon (n)=\begin{Bmatrix} 1 : n\equiv 1,4(mod 5)\\ -1 : n\equiv 2,3(mod 5) \end{Bmatrix}$ It is known that it is multiplicative, i.e $\epsilon(nm) = ...
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1answer
65 views

Why aren't definitions well formed formulas?

Why aren't definitions well formed formulas? For instance, the definition of an additive inverse is: "Let $x \in \Bbb Z$. Then the additive inverse of $x$ is $y \in \Bbb Z$ such that $x+y=0$". Why ...
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1answer
124 views

Proof that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too

I'm trying to figure out how to proof that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too. Can someone help me please?
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3answers
26 views

Absolute Value Inequality Including Itself

Given a real number $a$. Will it be correct to use the following inequality in the proof: $$-a\le|a|\le a$$ Although "less" and "greater" parts never actually happen, the whole equation will always ...
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1answer
28 views

Single variable calculus: confusion about directional derivative

I am stuck on deriving proving the following equalities: given $x,y,t$ scalar and $f$ a scalar function (1) $\lim\limits_{t \to 0} \dfrac{f(x + t(y-x)) - f(x)}{t} = \left.\dfrac{df}{dt}\right|_{t \to ...
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1answer
94 views

Is this interpretation of the Dirac-measure property correct?

First and foremost, apologies in advance for using an abuse of notation by placing the Dirac measure inside an integral for which I was told that this should not be done from a previous question asked ...
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2answers
41 views

Proving that $4000(1 - 0.95^n) $ is true for this situation

I can see why the following formula is correct, but I'm not sure how to set about proving it. A man needs to spread 4000kg of sand over his garden. He decides to spread 200kg every day, but after the ...
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1answer
101 views

Proofs involving triangles and rectangles

The figure below represents a rectangle ACLK with an inscribed right triangle ABC. The lower case letters represent lengths of segments (ex. x=|KB|, etc. a.) prove that triangle ABC is similar to ...
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2answers
29 views

Prove the following using induction

$$\frac{1}{1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}$$ I'm new to induction, but this is what I cam up with so far. $$1 - \frac{1}{k(k+1)} + \frac{1}{(k+1)(k+2)} = 1 - ...
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1answer
58 views

Does the empty set satisfy this statement?

Let K be the subset of |R (real numbers: Statement: John likes K if and only if ∃a∈K such that ∀x∈K, a < x Question: Does John like any subset of the real numbers? My answer: John will not ...
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2answers
44 views

Please explain this basic proof

In my freshman math course book there's a proof of associativity of addition on the natural numbers using mathematical induction. The author proves the base case and assumes the hypothesis, $a+(b+c) = ...
2
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1answer
101 views

Starting index of a sequence is irrelevant

"Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $c$ be a real number, and let $k \geq 0$ be a non-negative integer. Show that $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
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1answer
101 views

GCD induction proof

I apologize if this is a duplicate question (believe me, I've searched). This question is a part of an ungraded class warm-up exercise. Question: Using induction, prove that for all positive integers ...
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0answers
64 views

Can't prove this equation [closed]

I can not prove that $a = b = c = n = 0$ is the only answer of $5n^2 = 36a^2 + 18b^2 + 6c^2$ when $a, b, c, n$ are integers Please help me.
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1answer
28 views

Vector Space - Removal from both sides

Given $V$ a vector space, and $A$, $B$, $C$ as it's members, How can I prove this: $$A + B = A + C \implies B = C$$ I'm sorry for my Mathematical English. I don't know the exact ...
2
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1answer
36 views

If $\mathbb{Q}$ is dense in an (Archimedean) ordered field K, is K a complete ordered field?

Let K be an ordered field. Then K contains the smallest ordered filed $\mathbb{Q}$. If $\mathbb{Q}$ is dense and proper in K, is K a complete ordered field? If $\mathbb{Q}$ is dense and proper in K, ...
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2answers
181 views

Sum of independent symmetric random variables is symmetric

This is exercise 3.2.5 from Probability and Random Processes by Grimmett and Stirzaker: Let $X_r$, $1 \leq r \leq n$, be independent random variables which are discrete and symmetric about $0$, that ...
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4answers
876 views

Prove: if $x$ is even, then $x + 5$ is odd. [duplicate]

I am trying to prove or disprove that if $x$ is even, then $x + 5$ is odd. This is what I have thus far, but I am stuck: Assume that the chose variable (x) are in the domain: (x) is an integer ...
3
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1answer
53 views

Could someone make the proof into a hinted exercise?

Let $(c_n)$ be a sequence of positive numbers. Could someone make the proof of the inequality $\displaystyle\limsup_{n\to\infty}\sqrt[n]{c_n}\leq\limsup_{n\to\infty}\frac{c_{n+1}}{c_n}$ into a ...
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3answers
33 views

Need Help In Proving Exact Power of Prime Divides Product Of Factorials

I am trying to solve the following question: Prove that if a and b are positive integers, p is prime, and a + b = 2p - 1 then p || a!b!. Where || means that no higher power of p will divide a!b!. ...
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2answers
56 views

Proving $2^{n+1} < n^2 + 2$ for $n\geq 0$ by induction

I'm trying to prove that $2^{n+1} < n^2 + 2$ for $n \ge 0$ by use of mathematical induction, but I get to the inductive step and get lost. I don't know how to link my assumption to the proof.
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1answer
53 views

Prove derivative with summation by induction

I have this math question. That I am stuck on. If $f$ is a function, let $Df$ be its derivative. For $n\in \mathbb{Z}^+$ let $$ f^{(n)} = \underbrace{D \cdots D }_{n\mathrm{\ times}} f $$ be ...
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1answer
50 views

Prove that a set is a subgroup of $S_G$ (the set of all permutations of a group G)

The question: Suppose that $G$ is a group, and $\forall a\in G$, $f_a:G\to G$ is defined as $f_a(x)=ax, \forall x\in G$. If $S_G$ is the set of all permutations of G, prove that $H=\left\{ f_a:a\in ...
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1answer
26 views

Prove formula for $n^{th}$ derivative

I have this math question. That I am stuck on. If $f$ is a function, let $Df$ be its derivative. For $n\in \mathbb{Z}^+$ let $$ f^{(n)} = \underbrace{D \cdots D }_{n\mathrm{\ times}} f $$ be ...
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1answer
48 views

If any bounded sequence in an ordered field K has a subsequential limit in K, is K a complete ordered field?

Let K be an ordered field. If any bounded sequence in K has a subsequential limit in K, is K a complete ordered field? (i.e. satisfying any one of the equivalent definitions of a complete ordered ...
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2answers
30 views

Showing proof using contrapositive

Tell if the statement is true or false. If true provide a proof. $\forall x$ $\in R$ $\left(\forall M > 1 \left(x \ge 1-\frac{1}{M}\right) \to x \ge 1 \right)$ I believe this statement is True. ...
3
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3answers
87 views

Proof that in a metric space $X$, if $\phi \in \mathbb{R}^X$ is not continuous, then $\{ x \ | \ \phi(x) \geq \alpha \}$ is not necessarily closed

In the last few days I already posted two alternative proofs (here and the other available link) of the basic result in metric spaces that, given a continuous function $\phi \in \mathbb{R}^X$, the set ...
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2answers
39 views

Prove using the binomial theorem

I have this homework question: Let $n$ be a positive integer, and $\alpha$ any nonnegative real number. Use Binomial Theorem to show that $$(1+\alpha)^n\ge 1+n\alpha+\frac{n(n-1)}{2}\alpha^2$$ ...
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2answers
40 views

Prime numbers proof

My problem: Prove that a natural number $p$ is prime if and only if $p > 1$ and there exists no natural number $n \in \mathbb{N}$ with $1<n\le \sqrt{p}$ such that $n|p$. Help!
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2answers
44 views

How to prove $\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$ combinatorially

How can we prove combinatorially $$\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$$ I can get LHS by asking: How many ways can we form an $m+1$ person committee from a group of $n+1$ ...
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1answer
58 views

Choosing a $k$ person committee with chairperson from a group of $n$ people confusion

The following is from: http://www.math.sjsu.edu/~bremer/Teaching/Math163/Homework/HomeworkFiles/Solution03.pdf I am having trouble understanding these identities and the solutions. I am confused as ...
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0answers
23 views

Domain specification of derivative extension.

Given the definition of Taylor expansion: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ We can find the $m$'th derivative of $f(x)$ quite easily: $$\frac{d^m}{dx^m} f(x) = ...
0
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1answer
30 views

Proof based on convergence arguments that, if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \ | \ \phi (x) \leq \alpha \}$ is closed

Recently, I posted a proof of the proposition that, given a continuous function $\phi \in \mathbb{R}^X$, the set $\{ \ x \ | \ \phi (x) \geq \alpha \}$ is closed. Apparently, assuming that lack of ...