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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0
votes
2answers
31 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
31 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 ...
0
votes
3answers
52 views

Induction proof for natural numbers

I am unable to proceed with the below claim. $$2^{m} \times 2^{n} = 2^{m+n}$$ Could anyone let me know how to prove the above claim using induction proof? I was able to derive proof for odd natural ...
1
vote
2answers
20 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
44 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
1answer
399 views

proof of validity of tautology in first order logic

Every first-order logic formula which has a tautological shape in propositional logic is a valid formula. Will it be possible to give a formal proof for the above ? Thanks and Regards.
8
votes
4answers
680 views

Book for proof writing

Which of the two books is suited for a student looking to learn how to write proofs? I have a working knowledge of calculus and linear algebra but I'm not good at writing proofs. My intention is to ...
11
votes
8answers
866 views

Book about technical and academic writing

I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
21
votes
8answers
3k views

LaTeX/TeX Vs. Mathematica for Typesetting [on hold]

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regards to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that ...
1
vote
3answers
57 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 ...
1
vote
3answers
80 views

$f'(a)=0$ implies $x=a$ is not a simple zero of $f$

Let $a$ be the root of a polynomial $f(x)$ and let $f'(a)=0$. Then $x=a$ is not a simple zero of $f(x)$. What is the name of this theorem and does someone know a simple (high school level) proof?
0
votes
4answers
59 views

$|(0,1)| = |\mathbb R|$

For this problem in proving that the cardinality of (0,1) is equal to that of the set of real numbers, would I just prove that (0,1) is uncountable, and then use the theorem that the subset of an ...
2
votes
1answer
48 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), ...
2
votes
1answer
4k views

How can I simply prove that the pearson correlation coefficient is between -1 and 1?

For building a recommendation system, I also use the Pearson correlation coefficient. This is the definition: $r(x, y)=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n ...
2
votes
0answers
67 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
1
vote
2answers
56 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 ...
5
votes
2answers
78 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 ...
3
votes
1answer
56 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
vote
1answer
24 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
votes
1answer
19 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
0answers
29 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 ...
6
votes
1answer
70 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
3
votes
1answer
78 views

Is my Proof Correct and Rigorous: Proving that Quotient Space is Hausdorff

Question: Let $X$ be a topological space and let $A ⊂ X$. Define an equivalence relation $∼$ on $X$ such that the equivalence classes are: • $A$ itself, and, • Singletons {$x$} such that $x /∈ A$. ...
5
votes
3answers
295 views

How to prove or statements

How do I prove statements of the following types: $A \text{ or } B \implies$ C $A \implies B \text{ or } C$ I don't know how to go about proving statements like this when they have "or". Can ...
0
votes
0answers
38 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
1
vote
1answer
17 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
1answer
17 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
54 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
43 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 ...
3
votes
2answers
24 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
39 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 ...
3
votes
5answers
110 views

Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
0
votes
1answer
42 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. ...
-5
votes
1answer
37 views

Consider the expression n^5 + 9n. [closed]

a) Prove directly that $n^5 + 9n$ is even for all $n \in \Bbb N$ b) Prove by induction that $n^5 + 9n$ is divisible by $5$ for all $n \in \Bbb N$ c) Prove that for all $m \in \Bbb N$, $2 \mid m$ and ...
5
votes
1answer
110 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
-4
votes
0answers
29 views

Let $f: ℝ \to ℝ$ be the function given by $f(x)=2x-3$. Then $f$ is onto. [closed]

I need help proving or disproving this statement. Let $f: ℝ \to ℝ$ be the function given by $f(x)=2x-3$. Then $f$ is onto.
1
vote
1answer
16 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.
3
votes
2answers
75 views

$a_{n}$ converges and $\frac{a_{n}}{n+1}$ too?

I have a sequence $a_{n}$ which converges to $a$, then I have another sequence which is based on $a_{n}$: $b_{n}:=\frac{a_{n}}{n+1}$, now I have to show that $b_{n}$ also converges to $a$. My steps: ...
0
votes
1answer
34 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
2answers
30 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 ...
0
votes
1answer
23 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 $\frac {d}{dt}$?
-2
votes
2answers
45 views

Please help prove this summation problem for me [closed]

Prove that for all integers n greater than or equal to 1, $\sum_{k=1}^{3n} (4k+3)=3n(6n+5)$.
1
vote
1answer
36 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
30 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
44 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: ...
7
votes
2answers
54 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
2answers
40 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
1answer
132 views

Nonlinear partial differential equations with applications

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
1
vote
1answer
25 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
29 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 ...