For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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1answer
35 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
2
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2answers
81 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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1answer
52 views

Proof, that $a \equiv 1 \pmod{p}$

Let $n \in \mathbb{N}^{+} \smallsetminus \{{1}\}$ and $p = min\{p \in \mathbb{P} : p \mid n\}$. Also, let $a \in \mathbb{Z}$ and $a^n \equiv 1 \pmod{n}$ I need to proof, that $a \equiv 1 \pmod{p}$. ...
0
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1answer
39 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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2answers
167 views

Show a limit of piecewise function does not exist as x tends to 0

So I am given a function, f(x), as follows: f(x)=sin(1/x) if $0<x\leq1$ and f(x)=4 if x=0. clearly this is an example of a function integrable on [0,1] but it is discontinuous at x=0. How ...
0
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1answer
30 views

induction with factorials

I need help with this please. I understand step one is to let $n=1$. step two let $ n = k$. Step three prove for $k+1$. But I would like a clear example of each... Prove $$\sum_{i=1}^n ...
0
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1answer
34 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
1
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1answer
80 views

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$

Consider for real $a<b$ and real $A,B,C$, the function $f:[a,b] \to \mathbb R$ defined by $$f(x) = \begin{cases} A & x = a \\ B & x=b \\ C & a < x < b \end{cases}$$ I want to ...
0
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1answer
89 views

Prove the infimum and supremum of the positive rational numbers

I am having this set: $$ X= \mathbb{Q^+} = \{x \in \mathbb{R} \ \ |x \in \mathbb{Q} \ \text{and} \ x>0 \} $$ How can I prove that $\inf X= 0$ and there is no supremum ? (I think there is no ...
0
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1answer
35 views

Prove by contrapositive: Φ∪{β} ⊨ α & Φ∪{¬β} ⊨ α iff Φ ⊨ α

We are to prove this by contrapositive (by the way: Φ is a set of formulas of predicate logic and α a formula of predicate logic) I've managed the Right to Left proof, but I struggle with the Left to ...
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2answers
54 views

Division algorithm and Prime Numbers

In my class, the professor went through a proof that if $p|xy$ then $p|x$ or $p|y$. where p is a prime number. And now that I am reading through it, there is a small piece of the proof that I do not ...
3
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2answers
49 views

Prove that the sequence $a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded.

Prove that the sequence $a_{0} = \frac{1}{2}, a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded. Assume that $0 < a_{n} < 1$ for every $n$ and $a_{0} = \frac{1}{2}$. Prof. used induction to prove that ...
0
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1answer
25 views

Showing a function is a bijection without a specific function

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). Let ...
0
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2answers
53 views

Prove $f(a + b) = f(a) + f(b)$ in an ordered field

Let $F$ be an ordered field with identities $0$ and $1'$, and define $f: \mathbb{N} \to F$ by: $f(1) = 1'$, $f(x + 1) = f(x) + 1'$ (the addition for the right hand-side is addition in the field). So ...
0
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2answers
103 views

Pairs of consecutive values assigned to objects of different state.

Seven objects of state A and eight objects of state B are assigned values across a variant range of 15. What is the expected number of pairs of such consecutive values that are assigned to objects of ...
3
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1answer
30 views

Is this exercise right, or something is wrong or missing.

I have to find the following limit For each positive integer $n$ define: $$a_n = \frac{1}{n}\left[\left(\frac{1}{n}\right)^2 + \left(\frac{2}{n}\right)^2 + ... + \left(\frac{n}{n}\right)^2 ...
3
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0answers
33 views

Validity of this geometry proof

In triangle ABC, D is the midpoint of AB and E is the point of trisection of BC nearer to C. Given that $\angle$ADC = $\angle$ BAE find $\angle$BAC. Source Q5 Lets call the ...
2
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2answers
30 views

Prove that for any integers x,y there are integers a,b such that gcd(x,y) = ax + by

How would I go about proving that: For any integers x,y there are integers a,b such that gcd(x,y) = ax + by? One thing I noticed is that when x is a multiple of y or vice versa, the smaller number is ...
0
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0answers
46 views

Poker Hand Equivalent Relation

Let $P$ be the set of all possible poker hands. Define a relation $J$ of $P$ by $a$ is $J$-related to $b$ iff $a$ and $b$ have no cards in common. Is $J$ reflexive? Symmetric? Transitive? Having a ...
3
votes
1answer
51 views

If $A \subseteq B$, $a \in A$ and $a \not\in B \setminus C$, then $a\in C$

This is the question: Suppose that $A \subseteq B$, $a \in A$ and $a \not\in B \setminus C$. Prove by the method of contradiction that a ∈ C. This is my proof: Suppose by contradiction $a ...
2
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1answer
25 views

Is this a correct proof?

This is the question: Suppose x and y are real numbers, and 3x + 2y ≤ 5. Prove that if x > 1 then y < 1. I have tried to prove it in this way: Our goal is to prove that $y < 1$. We ...
0
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1answer
22 views

How can I prove this $x \neq 1$ $\rightarrow$ $\frac{y + 1} {y - 2} = x$

Why this solution is not a proof or is incorrect, and how can I solve it? $\frac{y + 1} {y - 2} = x$, where $y \neq 2$ So we have: $y + 1$ = $x(y - 2)$ $y + 1 = xy - 2x$ $y - xy = -1 - 2x$ $y(1 ...
2
votes
1answer
35 views

Disquisitiones Arithmaticae art. 49

Article 49. If $p$ is a prime number tat does not divide $a$, and if $a^t$ is the lowest power of $a$ that is congruent to unity relative to the modulus $p$, the exponent $t$ will either be $p-1$ or a ...
1
vote
1answer
41 views

Eigenvalues, polynomials and minimal polynomials

I have proved (a) by: Let $\lambda$ be an eigenvalue of $AB$ $ABv=\lambda*v$ Then $BABv=\lambda*B*v$ so Bv is an eigenvector of BA with eigenvalue $\lambda$. For B, I have found the formula in ...
0
votes
1answer
14 views

How to Simplify this question?

$$\begin{align}f(n+1) &= (n+2)! -1 = (n+2)(n+1)! - 1 \\ &= (n+2)\left((n+1)!-1\right) + (n+2) - 1 \\ &= (n+2) \cdot f(n) + (n+1) \end{align}$$ I understand the first line but not how to ...
1
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0answers
32 views

Application of existence and uniqueness theorem

Let $y_1,y_2$ be solutions on $(a,b)$ to $y'''+P(x)y''+Q(x)y'+R(x)y=0,$ where $P,Q,R$ are continuous functions on $(a,b).$ Let $x_o \in (a,b).$ Suppose $y_1(x_o)=y_2(x_o)=0$ and ...
3
votes
1answer
38 views

Prove that if a fraction is broken up into two the resulting two fractions cannot both have a larger value [duplicate]

I am working on research involving probability tables. I simplified the problem to the following. Say we have the following: $C_1, C_2, C_3, C_4$ $\forall i, C_i > 0$ $x = \frac{C_1 + C_2}{C_1 + ...
2
votes
2answers
28 views

Showing that a function $f$ on some interval $I$ is a contractive function

The sequence $(u_{n})_{n\geq1}$ is recursively defined by : $u_{1} = α$ $ ϵ $ $ [0, 16]$ $u_{n+1} = \sqrt{20 - u_{n}} $ ,for every natural number $n$ Show that the function $f(u) = \sqrt{20 ...
1
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0answers
42 views

How does one move from step to step in a math proof?

I initially can set up a proof(I can write out the givens and what is need to prove) but I struggle moving line to line. How do you overcome this and what strategies do you use?
1
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2answers
27 views

Find the limit of a function using the definition

I want to prove the limit of the function sqrt(x) equals 2 using the definition of the limit of a function. The definition of the limit of a function is: Let I be an open interval that contains the ...
3
votes
0answers
102 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
0
votes
3answers
24 views

Crossed induction

I have two sequences of equalities $L_n$ and $R_n$. The equation $L_{n+1}$ is true only if $R_n$ is true, and the same happens for $R_{n+1}$ $$R_n \implies L_{n+1}$$ $$L_n \implies R_{n+1}$$ How ...
0
votes
1answer
37 views

General solution of Cauchy -Euler equation

Consider $x^2y''+\frac{5y}{4}=0, \ x>0.$ The general solution is: $c_1x^{1/2}\text{cos}(\text{ln}x)+c_2x^{1/2}\text{sin}(\text{ln}x), \ x>0$ Could anyone advise me how to show ...
1
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1answer
44 views

Prove a predicate formula in the constructive logic

Using the constructive logic (the axiom $A\lor\lnot A$ cannot be used), using quantifier axioms and Modus Ponens, and Generalization, prove the following: $\exists x(B(x) \to C(x)) \to (\forall xB(x) ...
1
vote
3answers
32 views

Estimations in a ordered field?

My Problem: I am stuck with a proof strategy on the following: So i have got an ordered field $ (K,+,*,<) $ given. I also have $x,y\in K$ and $0\le y < x$ I have to proof that, for every n ...
2
votes
1answer
49 views

If $f\colon (0,1)\to(0,1)$ is surjective, is there necessarily an ordered (incr. or decr.) subsequence $\{x_i\}$ in the domain st $f(x_i)\to1$?

Let $f\colon (0,1)\to(0,1)$; $f$ is surjective. The claim is that, for $f$, there is either a strictly increasing or decreasing subsequence of $x_i$ in the domain for which $f(x_i)$ approaches $1$. ...
0
votes
2answers
35 views

$A\neq \varnothing $, $B\neq \varnothing$, $A\neq B.$ Prove $A\times B \neq B\times A$ [closed]

$$A\neq \varnothing ,B\neq \varnothing,A\neq B. \\\text{Prove }A\times B \neq B\times A$$ I'm pretty sure this has to do with inverse for relations. But I'm not sure how to begin proofing ...
5
votes
2answers
83 views

Intuition behind sum of multiplication arithmetic sequence

Maybe this is a stupid question but please guide and enlighten me patiently. I have just known something fact that quite shocking me. Let start from this simple fact $$\sum_{k=1}^n ...
2
votes
0answers
57 views

A Proof by Induction about terms and variable assignments

I am (sort of) familiar with inductive proofs about wffs, but proofs by induction about terms took me by surprise. Prove by Induction that: if variable assignments q, q' agree on all variables ...
6
votes
2answers
126 views

Why is this proof that $\sqrt{2}$ is irrational titled as “Proof by infinite descent”?

I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of infinite descent. I understand it as: We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are ...
1
vote
1answer
30 views

Continuous $f:\overline{\mathbb{D}} \to \mathbb{C}$ such that $f$ is holomorphic in $\mathbb{D}$ and $|f(z)|=1, \forall z \in \partial\mathbb{D} \ ?$

Could anyone advise me how to find all continuous $f:\overline{\mathbb{D}} \to \mathbb{C}$ such that $f$ is holomorphic in $\mathbb{D}$ and $|f(z)|=1, \forall z \in \partial\mathbb{D} \ ?$ Do I use ...
1
vote
1answer
157 views

supremum is the only positive root of $z^n=a^m$

I'm trying to define $a^x$ with $x$ rational. Let $r=\frac{m}{n}$ with $m,n \in \mathbb{Z}$, and let $a>1$ be a real number. We define $S_{r}(a)= \left \{ x \in \mathbb{R} | 0\leq x^n\leq a^m ...
1
vote
2answers
48 views

Proof By Induction Fibonacci Numbers

How do I prove that $$ f_{ 2n+1 } = 3f_{ 2n } + 1 - f_{ 2n-3 } $$ I'm not sure how to prove it using the defining recurrence of Fibonacci numbers.
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2answers
57 views

Logical Proofs involving powersets [duplicate]

I have no idea how to work on the following proof. Any Suggestions? Prove that for any sets A and B, if P(A) ∪ P(B) = P(A ∪ B) then either A ⊆ B or B ⊆ A. Thanks
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votes
5answers
83 views

Series and sequences convergence with a certain condition.

Let $\sum_{n=1}^\infty (a_{n})$ converge. Let $\{n_{k}\}$ be a subsequence of the sequence of positive integers. For each $k$ define $b_{k}=a_{n_{k-1}+1}+...+a_{n_{k}}$ where $n_{0}=0$. Prove that ...
0
votes
2answers
36 views

Proving that a certain series converge if and only if the a_n converges

I need to prove the following statement: Let $\{a_n\}$ be a sequence of real numbers.Prove that $\sum_{n=1}^\infty (a_{n}-a_{n+1})$ Converges iff $\{a_n\}$ converges. If $\sum_{n=1}^\infty ...
0
votes
2answers
54 views

Prove the convergence of a sequence involving integrals

I need to prove the following: Assume $f:[a,b] \to R$ is continous, $f(x)\leq0$ for all $x \in [a,b]$, and $M=sup\{f(x):x \in [a,b]\}$.Show that: $$\{[\int [f(x)]^{n}dx]^{1/n}\} \to M$$ This result ...
0
votes
1answer
31 views

“Natural” Homeomorphisms, Retracts and Knots

I have been trying to prove the following result for a few days now, and have made some amount of progress, but now I'm struggling. This is what I'd like to prove: Every proper knot, K has a retract ...
0
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0answers
43 views

Why is this kernel isomorphic to $\Omega^{n-2}(E-2D)$?

I'm reading this article and I didn't understand the proof of the item (1) of this proposition on page 225 (see below): I have the following questions: I didn't understand how these inclusions ...
7
votes
2answers
109 views

An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$

How does one prove the given series? $$\sum_{n=0}^\infty\left(\frac{(-1)^n}{4n-a+2}+\frac{(-1)^n}{4n+a+2}\right)=\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$$ This series came up in xpaul's ...