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
32 views

Verify my proof: if $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive.

Could someone verify my proof and my writing? Proposition: If $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive. Definition 1: A binary ...
0
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1answer
9 views

question about vacuous truth and function

I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of ...
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7answers
50 views

Induction proof of $1 + 6 + 11 +\cdots + (5n-4)=n(5n-3)/2$

I need help getting started with this proof. Prove using mathematical induction. $$ 1 + 6 + 11 + \cdots + (5n-4)=n(5n-3)/2 $$ $$ n=1,2,3,... $$ I know for my basis step I need to set $n=1$ but I ...
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votes
2answers
49 views

If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$

Suppose that $k$, $n$, and $d$ are integers and $d$ is not $0$. Prove: If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$. You may not use the theorem stating the following: Let ...
1
vote
1answer
44 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...
2
votes
0answers
19 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
2
votes
0answers
70 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
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3answers
83 views

Real Analysis Proofs

I am taking a Real Analysis class using the textbook Analysis with an Introduction to Proofs, $5^{th}$ Ed. by Steven Lay. So far I am not understanding the proofs at all. Does anyone know of any good ...
2
votes
2answers
80 views

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction [duplicate]

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction My proof so far: Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ Base Case $P(1):$ LHS = $1^3 = 1$ ...
2
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0answers
56 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
0
votes
1answer
49 views

Prove that $nCr = n(n-1)(n-2)\cdots(n-r+1)/ 1\cdot2\cdot3 \cdots r$ is an integer for all positive integral $n$ and for all integers $r \geq 0$.

Prove that $nCr =\frac{ n(n-1)(n-2)\cdots(n-r+1)}{ 1\cdot2\cdot3 \cdots r}$, is an integer for all positive integral values of $n$ and for all integers $r \geq 0$. Can someone please explain it to ...
-1
votes
2answers
62 views

Inequality $\prod\limits_{r=1}^{- \infty}(1+(\frac{1}{2})^r)<\frac 52$ [closed]

Prove this inequality. $\prod\limits_{r=1}^{- \infty}\left(1+\left(\frac{1}{2}\right)^r\right)<\dfrac 52$ I have tried to prove it using induction but it is not coming.
2
votes
1answer
81 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
0
votes
2answers
77 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
1
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2answers
31 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
votes
1answer
59 views

Attempt to proof the Cantor-Bernstein theorem

I've found a proof of the Cantor-Bernstein theorem in Kleene's 'Introduction to Metamathematics' (1952) in §4 Thm A. I must admit I don't understand its essence but I was wondering if the proof could ...
1
vote
1answer
25 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
1
vote
1answer
35 views

How to display one to one correspondence?

This is a problem from Discrete Mathematics and its Applications Here is the book's definition of countable/not countable For 2a, I came up with the fact that the set is countably infinite. What ...
2
votes
2answers
44 views

Prove that $ A \subseteq B \iff \mathcal{P}(A) \subseteq \mathcal{P}(B) $.

I'm going through Velleman's How To Prove It and I'm currently on section 3.4 which deals with techniques for proofs involving conjunctions and biconditionals. The title of this question is from one ...
0
votes
1answer
31 views

Proof Using iff Intermediate Lines

I am posting this question motivated by Bungo's response to my question here -- scroll down to his/her response and comment. It was the first time I've seen this technique. It looks like a circular ...
0
votes
2answers
30 views

Uniform convergence $f_n(a) = a^{4n} + \frac1{n^2}$

I have $f_n(a) = a^{4n} + \frac1{n^2}$ which I know converges to $f(a)=0$ uniform on theinterval $[0,1)$ This works? $\lim \limits_{n \to \infty} a^{4n} + \frac1{n^2} = \lim \limits_{n\to\infty} ...
0
votes
2answers
72 views

Proof of a point's existence in an open interval

Well let us begin consider a set $A$ $$A = \{a \le x \le b \space | \space f(x) > 0 \}$$ Lets take $\alpha = \sup A$ and $\beta = \inf A$ What we must do is prove that $\alpha = d$ and ...
1
vote
1answer
25 views

Next step to take in direct proof for one to one?

This is from Discrete Mathematics and its Applications And the definition of strictly increasing. Here is my work so far. I know that a direct proof involves making an assumption p, which in ...
0
votes
0answers
19 views

How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular ...
1
vote
1answer
89 views

How to prove that $\cos(\pi÷11)+\cos(3\pi÷11)+\cos(5\pi÷11)+\cos(7\pi÷11)+\cos(9\pi÷11)=0.5$? [duplicate]

I need to prove that $$\cos\dfrac{\pi}{11}+\cos\dfrac{3\pi}{11}+\cos\dfrac{5\pi}{11}+\cos\dfrac{7\pi}{11}+\cos\dfrac{9\pi}{11}=\dfrac{1}{2}$$ How to do it?
2
votes
4answers
74 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
2
votes
2answers
72 views

Proving by induction that any two natural numbers are equal.

This is something I've been working on for a while now; although it seems trivial, I am confused. I can't seem to find the error. Originally I thought the problem was with the base case, then I ...
2
votes
1answer
36 views

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous My counter example is $\begin{cases} 1 - x & \text{ if } -2 \leq x \leq 1 \\ 2 - x & \text{ if } 1 < x \leq 2 ...
2
votes
2answers
30 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
1
vote
1answer
12 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
0
votes
1answer
15 views

Finding a bijective function from $\prod_{i\in I}X_i$ to $\bigl(\prod_{j\in J}X_j\bigr)\times\bigl(\prod_{k\in K}X_k\bigr)$

If $(X_i)_{i\in I}$ is a family of sets and $J,K$ are non-empty disjoint sets of $I$ such that $I=J\cup K$, then show that there is a bijective function from $\prod_{i\in I}X_i$ to ...
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votes
2answers
41 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
0
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4answers
98 views

Why can't I prove this statement by simple induction? Sum of $1/2^1 + \cdots+ n/2^n = x$

I have to prove the following: $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{2 + n}{2^n}. $$ I am trying to prove this by simple induction. First, I proved that $P(1)$ ...
1
vote
1answer
40 views

How to proove the following general form of proof

Suppose I have a statement $p(m,n)$ where $m,n \in \mathbb{N}$ that I want to proove. Suppose I have proofs of the following: $p(1,n)$ true for all $n \in \mathbb{N}$. $p(m,1)$ true for all $m \in ...
0
votes
1answer
25 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
0
votes
1answer
50 views

Let ${a_n},{b_n}>0 ,\lim \limits_{n \to \infty} [a_n+b_n]=0 $ then$ \lim \limits_{n \to \infty}a_n=0 $ and$ \lim \limits_{n \to \infty} b_n = 0$

Let ${a_n}$ and ${b_n}$ be sequences of nonnegative numbers. Show that if $\lim \limits_{n \to \infty} [a_n+b_n]=0$ then $\lim \limits_{n \to \infty}a_n=0$ and $\lim \limits_{n \to \infty} b_n = 0$. ...
0
votes
1answer
52 views

Property of continuous functions regarding maximum

Claim 1: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ assumes a maximum value I know there's a theorem that states if $f$ is a continuous real-valued function on a closed interval ...
-1
votes
2answers
89 views

How to show that “Uniformly continuous implies continuous”? [closed]

Can I go from the definition of uniformly continuity to continuity? Please somebody show me how to do that. Thanks.
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2answers
37 views

How to find all the equivalence classes for a specific equivalence relation?

What are the equivalence classes of the following equivalence relation $$S=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x - y \in \mathbb{Q} \}$$? I know that an equivalence relation $R$ on a set ...
0
votes
1answer
43 views

Limit of $x^2+3\sin x$ as $x$ goes to negative infinity [closed]

For every $x, x^2+3 \sin x \ge x^2 −3$ and, for every $c \ge 3, c^2 −3 \ge c$ hence, defining $x(c)=−c,$ one gets: $\forall c\ge 3, \exists x(c), \forall x \le x(c), x^2 + 3\sin x \ge c.$ I got this ...
0
votes
1answer
14 views

If there are $n$ $Y$'s for every $X$, but $m$ $X$'s for every $Y$, then the total number of $Y$'s is $n/m$ times the number of $X$'s?

(Apologies for the lengthy title. If you can make it shorter but still descriptive, please do so.) There is a certain form of reasoning that I find myself using every once in a while. I believe it's ...
0
votes
0answers
28 views

proving prime number's divisors

Let p ̸= 0, ±1 be an integer. Prove that p is prime if and only if p satisfies the following property: Whenever a and b are integers such that p = a · b, either a = ±1 or b = ±1. I proved the forward ...
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1answer
22 views

$S$ is a reflexive closure of $R$. Prove that if $R$ is symmetric, then so is $S$

Suppose $R$ is a relation on $A$, and let $S$ be the reflexive closure of $R$. Prove that if $R$ is symmetric, then so is $S$. S is the reflexive closure of $R$, which means that $$\forall x ...
0
votes
1answer
27 views

If $R_1$ and $R_2$ are symmetric and transitive, prove that also $R_1 \cup R_2$ is symmetric and transitive

I have to prove or confute (with a counter example) that, if $R_1$ and $R_2$ are symmetric and transitive, then $R_1 \cup R_2$ is also symmetric and transitive. Note: To prove the transitive case, ...
2
votes
2answers
94 views

If n^2 is even n is even

I understand that there are already several answers to how to prove this question: Prove if $n^2$ is even, then $n$ is even. Prove that if $n^2$ is even then $n$ is even I am trying to understand ...
2
votes
1answer
47 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
0
votes
2answers
38 views

What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...
0
votes
1answer
37 views

Next step to take in direct proof?

This is a problem from Discrete Mathematics and its Applications. I understand the basic ideas of the direct proof. Basically a proof is a conclusion from a series of steps to establish the truth of ...
2
votes
1answer
44 views

Next step to take in direct proof or a workaround around current dilemma?

This is a problem from Discrete Math and Its Applications I used a direct proof to do this proof. I understand the process/idea behind the direct proof, mainly (from ...
2
votes
1answer
59 views

$\exists x (P(x) \to \forall y P(y))$ [duplicate]

Prove $\exists x (P(x) \to \forall y P(y))$. Let x = y. Suppose P(x) is true. Let y be arbitrary. Since P(x) is true, it must be that P(y) is true. Since y was arbitrary, we can conclude that ...