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

0
votes
2answers
59 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
50 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
50 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
64 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
112 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
28 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
101 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
99 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
74 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
66 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
75 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
17 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 ...
6
votes
2answers
90 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
67 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
37 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
20 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
54 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
56 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
45 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. ...
3
votes
4answers
82 views

Proof of divisibility using modular arithmetic: $5\mid 6^n - 5n + 4$

Prove that: $$6^n - 5n + 4 \space \text{is divisible by 5 for} \space n\ge1$$ Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using ...
-1
votes
1answer
36 views

Proof that this set is convex set

I need a help with prooving that a given set is a convex set: $A \in \mathbb{R}^{m\times n}, c \in \mathbb{R}^n , b\in \mathbb{R}^m:$ $? \in argmin\{ c^Tx| Ax=b,x \geq 0\}$ I know the definition of ...
0
votes
1answer
46 views

$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem Let $x,y,z \in ...
3
votes
3answers
44 views

If $f$ continuous in $[-1,1]$ Then $g(x)= \int_{-x}^x \! \, f(t)dt$ derivative in $[-1,1]$ and $g'(x)=f(x)+f(-x)$

I have this problem its a proof/disproof problem. For some reason I get wrong answer. If $f$ continuous in $[-1,1]$ Then $g(x)= \int_{-x}^x \! \, f(t)dt$ derivative in $[-1,1]$ and ...
3
votes
1answer
111 views

$x^n y^n = (xy)^n$, proof exercise

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem \begin{equation*} x^n ...
0
votes
0answers
31 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
0
votes
3answers
44 views

Is this a valid sum formula for rational functions?

Consider: $$\frac{a(x)}{\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}x^{2n-2}}$$ Is the expressions above equivalent to: $$\sum_{n=1}^{\infty} \frac{a(x)}{(-1)^{n-1}x^{2n-2}}$$ ??
5
votes
3answers
186 views

Prove that the real root of $x^3 + x + 1$ is irrational

Using wolframalpha.com we get that the real root of this polynomial is $-0.68233$ The only way that I have found how to prove it is using the Rational Root theorem. Using that theorem the possible ...
5
votes
2answers
121 views

$P(x)=x^3+ax^2+bx+c$, Proof $e^{P(x)}=\sin x$ has a solution.

Let $P(x)=x^3+ax^2+bx+c$ Proof : $e^{P(x)}=\sin x$ has a solution. I thought about it, and still cannot find where to start. Any ideas?, Thanks!
0
votes
0answers
49 views

Spivak proof for Polynomial existence of a root.

Spivak is proving that a odd function $f(x)$ has atleast one root, I almost understand, I just need a little help. The part I dont understand is $(*)$?? I see why he does so that the inequality ...
0
votes
1answer
59 views

Proof that $\{x \}$ is nowhere dense if and only if $x$ is not an isolated point of $X$.

The following is a proof of the result in the title of the question. I am writing it for various reasons, in particular to check if the proof is indeed correct, and to get a feedback on my writing ...
1
vote
2answers
49 views

Set theory proof problem about bounds [duplicate]

Suppose $A \ne \emptyset$ is bounded below. Let $-A$ denote the set of all $-x$ for $x$ in $A$. Prove that $-A \ne \emptyset$ that $-A$ is bounded above, and that $-\sup(-A)$ is the greatest lower ...
3
votes
0answers
64 views

Problem on symmetric polynomials

The following problem is from "Analysis I" by Amann/Escher. Exercise: There are obvious operations of $S_m$ on $\mathbb{N}^m$ and on $R[X_1,\dots,X_m]$. A polynomial $p\in R[X_1,\dots,X_m]$ is called ...
1
vote
4answers
45 views

Convergent sequence: the step “for n > N”

Below is a proof from the book "How to think about Analysis" by Lara Alcock. I'm bad at proofs, but working on it. So in the proof below I miss the reasoning for the line in the red rectangular. ...
-1
votes
1answer
40 views

How to prove this GCD theorem [closed]

I'm trying to prove the following: Write $a/b$ as $kx/ky$, where a$, b, x,$ and $y$ are positive integers and $k$ represents the greatest common factor of $a$ and $b$. Then $\frac{x}{y} = ...
0
votes
2answers
46 views

How to prove a function is onto?

I know the basic concept of onto but I just don't get how do you prove is onto. I know that the range = co-domain for it to be onto but I just don't understand the proofs given. For example how would ...
1
vote
1answer
74 views

subsequence converges to L implies L is a limit point of sequence

Proposition: Let $(a_n)^\infty_0$ be a sequence of real numbers, and let $L$ be a real number. Then the following two statements are logically equivalent: (a) $L$ is a limit point of ...
1
vote
2answers
94 views

Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$

Prove that $[2x]+[2y] \geq [x]+[y]+[x+y]$ whenever $x$ and $y$ are real numbers. The $[]$ symbol is the greatest integer or floor function. I have proved this fact by cases, but I stumbled upon what ...
4
votes
2answers
155 views

Integral along a contour is $0$, how?

I recently had an extremely failed attempt at asking the same question, so I am posting the same question more or less to hope that someone can give me feedback. Consider the integral: ...
0
votes
0answers
28 views

Estimation Lemma when going to $0$

Here is the problem: Contour Integral problem With help from Jack D'Aurizio We were able to prove that the contour integral of the big semi circle $=0$ as $R \to \infty$. Now the problem is the ...
1
vote
1answer
79 views

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, ...
0
votes
2answers
43 views

Showing that the $\lim s_n\neq\dfrac{2}{3}$ when $s_n=\dfrac{2}{3n}$

I'm trying to verify if I what I did to show that limit does not exist is valid using the negation of the definition: $\exists \epsilon>0, \forall N \in \mathbb{N}$ such that $n>N \implies ...
3
votes
2answers
57 views

How do I prove the circumference of the Koch snowflake is divergent?

How do I prove that the circumference of the Koch snowflake is divergent? Let's say that the line in the first picture has a lenght of $3cm$. Since the middle part ($1cm$) gets replaced with a ...
1
vote
2answers
29 views

How to replace a complex term in an equation using a function?

I have recently been working on a few models that look at mosquito predation. Now one of the peers wants me to add the complete equation of my model in the manuscript. I previously had the equation ...
2
votes
1answer
64 views

$\dim V = \dim \phi(V)+\dim \ker \phi$

I want to show that $\dim V = \dim \phi(V)+\dim \ker \phi$. I know this proof can be found in any linear algebra textbook. However, my question is not exactly about the proof, but on a statement I ...
1
vote
3answers
140 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
5
votes
2answers
101 views

Closed form of a Definite Integral [duplicate]

I attempted to integrate the following function from a practice problem in my Calculus textbook: $$\displaystyle \int_{0}^{\frac{\pi}{2}}{\frac{1}{1+\tan^\sqrt{2}(x)}} \ {\rm d}x$$ I failed to find ...
1
vote
1answer
31 views

Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
2
votes
0answers
62 views

Derivative of a composition of function - nice proof

Let's consider the well known "fake" proof below for the derivative of the composition of functions: Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let ...
3
votes
1answer
78 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...