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.

learn more… | top users | synonyms

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
31 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
43 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
30 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
64 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
20 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
88 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
71 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
25 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
11 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
14 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 ...
0
votes
2answers
40 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
votes
4answers
97 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
39 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.
1
vote
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 ...
1
vote
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
91 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
37 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
43 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 ...
1
vote
2answers
102 views

How to calculate $\sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$ [closed]

How can I calculate the summation: $\displaystyle \sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$
0
votes
1answer
23 views

Cases for x in $ \forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x) $.

This is from Velleman p145, problem 28. Theorem: $\forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x)$. Author's Proof: Let x be an arbitrary real number. Case 1. $x = 0$. Let $y ...
0
votes
2answers
18 views

If $g\circ f$ is $1$-$1$ then $f$ is $1$-$1$ but $g$ is not necessarily $1$-$1$.

Let $f:X\longrightarrow Y$ and $g: Y\longrightarrow Z$. Show that, if $g\circ f$ is $1$-$1$, then $f$ is $1$-$1$, but $g$ is not necessarily $1$-$1$ I don't know how to start the proof. We have ...
2
votes
2answers
61 views

Must proofs always be cited (Thesis)?

I have some proofs of theorems in my thesis that are very similar to the proofs from the literature ( "my" proofs are more extended and have more explaination, the structure isn't the same either). ...
1
vote
2answers
85 views

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn't be formal, but in the "usual" ...
1
vote
1answer
48 views

Prove that a function is continuous at $x = x_{0}$ using the $\delta - \epsilon$ definition

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$ $\forall \epsilon > 0$, $\exists \delta = ?$ ...
0
votes
1answer
51 views

Prove that function $f$ is continuous at $x = x_{0}$

In class we're given the following definition about continuity, and I want to apply this definition to the problems that follow: $f$ is continuous at $x_{0} \in \mathrm{dom}(f)$ if $\forall x_{n} \in ...
2
votes
1answer
65 views

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$

Prove that $$f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$$ is continuous at $0$ and discontinuous everywhere else Proof: ...
0
votes
2answers
16 views

Why do I need to know that rng R = A for this proof?

Let $A$ be a nonempty set. Show that if $R$ is a symmetric and transitive relation on $A$ such that $rngR = A$, then $R$ is reflexive on $A$. So I proved this by saying: For all $x,y\in A$, $(x,y)\in ...
5
votes
2answers
55 views

How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
0
votes
1answer
45 views

$f(A\cap B)=f(A)\cap f(B)$. Where's the mistake?

I'm trying to prove something that is false, to see where is the contradiction. I want to prove that if $f:X\longrightarrow Y$ and $A,B\subseteq X$ then $f(A\cap B)=f(A)\cap f(B)$. So, let $y \in ...
1
vote
4answers
33 views

Proving the remainder when a polynomial is divided by an integer.

How should I go around proving that $\forall x \in \mathbb{Z}$, the remainder when $x^2+2x$ is divided by $3$ is $0$ or $2$? Do I use the division algorithm for this one?
0
votes
2answers
17 views

Proving using the definitions of “strictly dominated by” and “dominated by”

Let $A, B,$ and $C$ be sets. If $A$ is strictly dominated by $B$ and $B$ is dominated by $C$, then $A$ is strictly dominated by $C$. I need to prove this using the definitions of "dominated by" ...
1
vote
2answers
51 views

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous. I know that the Dirichlet function is discontinuous everywhere ...
0
votes
0answers
37 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
0
votes
1answer
32 views

A proof that the Cantor set is Perfect

I found in a book a proof that the Cantor Set $\Delta$ is perfect, however I would like to know if "my proof" does the job in the same way. Theorem: The Cantor Set $\Delta$ is perfect. ...