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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3
votes
3answers
37 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
97 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
28 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
42 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
173 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
116 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
43 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
42 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
42 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
58 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
35 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
35 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
30 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
44 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
58 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
149 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
25 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
65 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
42 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
45 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
27 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
50 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
64 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
89 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
26 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 ...
1
vote
0answers
38 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
59 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 ...
-1
votes
1answer
27 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
1
vote
1answer
22 views

Could you expand a little on this proof or Floyd-Warshall Algorithm?

I'm reading this. $\quad$ He gives a proof of Floyd-Warshall's algorithm but I don't understand what he's doing nor why it proves that. I can see an intuitive proof in my mind that is as ...
1
vote
2answers
75 views

How to find the limit of the sequence given by $X_{n+1} = \frac12 X_n + \frac1{X_n}$

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1{X_n}$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still ...
1
vote
1answer
21 views

Prove center of a group is a subgroup using one-step subgroup test

I'm not sure if this is correct. It doesn't seem so. If $a,b \in C$, then we must show $ab^{-1} \in C$. $$ab^{-1}x=axb^{-1}=xab^{-1}$$ This doesn't seem correct. I've seen two-step subgroup tests ...
2
votes
2answers
63 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
2
votes
3answers
47 views

Why can't a direct proof be made backwards?

Say we have the following implication: $$\textit{Let $x\in \mathbb{Z}$. If $5x-7$ is even, then x is odd. }$$ The method used by my book to prove this implication is by means of a proof by ...
2
votes
1answer
58 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
3
votes
2answers
26 views

Given a set $A$, how do I prove that there exists a set of all sets $x$ such that $\bigcup x=A$?

I am working with Zermelo-Fraenkel axioms. Specifically, I am allowed to assume the Axiom of Pair, Axiom Schema of Comprehension, Axiom of Union, and Axiom of Power Set, etc. (not yet allowed to use ...
1
vote
2answers
41 views

How to prove $x^{\phi(m)+1}\equiv x\pmod{p}$ [duplicate]

How do I prove that $x^{\phi(m)+1}\equiv x\pmod{p}$ when $m=pq$, two distinct primes? I kind of have an idea that it involves Euler's Theorem but it doesn't seem to be working as well as I wanted it ...
0
votes
1answer
53 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
1
vote
1answer
21 views

show that a loopless graph $G$ contains a bipartite spanning subgraph $H$ such that $d_H(v) \ge \frac{1}{2} d_G(v)$ for all v $\in$ V.

The hint in the appendix of book says that bipartite subgraph with with largest possible number of edges has such a property, but I don't know how to use this hint! any help would be appreciated.
0
votes
1answer
38 views

Calculus proof attempt

Why does $$\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}$$ not yield the same result as $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\dot{x}}\right) \ ...
0
votes
1answer
25 views

second derivative of a parametric equation

can someone please explain how in the proof for the second differential of a parametric function we get from to ? how do we calculate $\frac {d}{dt}$?
0
votes
2answers
33 views

Induction proof for $x \le y $

So these are $x \in \{1 ... i\}$ and $y \in \{ i + 1 ... n\}$, for $i, n \in \mathbb N$. I want to prove it for every $x \le y $. I know it`s easy but the solution is escaping me. I have tried with ...
1
vote
1answer
39 views

Prove or disprove this lemma for Catalan Numbers

Prove or disprove that for all non-negative integers $n$ and $r$ with $r+1$ is less than or equal to $n$, $C(n,r+1)=C(n,r)\times\frac{n-r}{r+1}$.
0
votes
0answers
33 views

How to prove that all powers of two minus one have only 1's when in binary representation?

It just came to my mind that all powers of two, when represented in binary format, are composed of only 1's, not 0's. I can see some logic behind it, however I can't seem to come up with an actual ...
2
votes
2answers
50 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
1
vote
2answers
41 views

How does this proof that the square of an integer, not divisible by 5, leaves a remainder of 1 or 4 when divided by 5 work?

Below I have a part of a proof of the fact that the square of an integer, not divisible by $5$, leaves a remainder of $1$ or $4$ when divided by $5$. But I am wondering where does the part highlighted ...
7
votes
2answers
60 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
1
vote
1answer
26 views

Proving if $\frac{3x+1}{x-1}$ is onto?

So, I have this function: $f(x)=\frac{3x+1}{x-1}$. So, in proving if it is onto, then by definition, for every b in B, there exists an a in A such that $f(a)=b$. So, let's solve or a. We get: ...
1
vote
1answer
32 views

How many square root matrices?

I would like to see a proof of the following statement: A positive-semidefinite matrix has precisely one positive-semidefinite square root, which can be called its principal square root. I ...
1
vote
1answer
45 views

Consider the function $g(x)=xe^x$. Make and prove a conjecture about the $n^\text{th}$ derivative of $g$. [closed]

Please help on this homework problem I have in my Proofs class. My prof is really bad at explaining and I don't know how to answer this problem! Thank you!
0
votes
3answers
67 views

Prove that a rational number minus an irrational number must be irrational. [duplicate]

Please help with this homework problem I have! I don't know how to prove this.