For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0
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1answer
27 views

Prove that the last process is responsible for $\mathrm\lceil {n\over p} \rceil$ elements

In my parallel computing class, I have this proof in one of my HW problems: Suppose we have chosen a block agglomeration of $\mathrm n$ elements (labeled $\mathrm 0, 1, ... , n - 1$) to $\mathrm p $ ...
0
votes
2answers
41 views

How to apply the mean-value theorem to prove

$f$ is a function that is continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose that there exists $c \in (a, b)$ such that $f(c) > f(a)$ and $f(c) > f(b)$. Prove that there exists $d ...
0
votes
0answers
46 views

Proof: The reduced row echelon form of a matrix is unique.

If $A \in M_{m\times n}$ with real entries, then there exist a unique matrix $R$ in row echelon form such that $A\sim R$, where $R$ comes from $A$ after performing elementary operations. How can I do ...
7
votes
3answers
102 views

Show that if $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective

Well my question this time is: How to show that $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective I was trying to consider the function $g(x,y)=(f(x,y),y)$, ...
1
vote
2answers
44 views

Little trouble with inner product space

I am having a little difficulty trying to solve a beginner proof in the topic of inner product spaces. The statement says, suppose $a_1,…,a_n \in \mathbb R$, Prove that $(a_1+…+a_n)^{2}/n \le ...
0
votes
1answer
49 views

Suppose that the function f:(a,b) --> R is uniformly continuous. Prove that f:(a,b)-->R is bounded.

Also a second unrelated question, suppose that the function $f:[0,1] \to \Bbb{R}$ is continuous with $f (0)>0$ and $f (1)=0$. Prove that there is a number $x_0 (0,1]$ such that $f (x_0)=0$ and ...
0
votes
2answers
41 views

Working on an inequality

Hey all I am trying to show that an inequality involving integrals is true, and I am not sure if I am taking the right approach or what I am missing. I will write one of the inequalities I am trying ...
0
votes
1answer
46 views

Prove compact sublevel sets imply coercivity

$f$ is convex and $dom(f):x \in \mathbb{R}^N$. Define sublevel sets of $f$ as \begin{equation} \mathbf{S}(f,\beta)=\{x \in \mathbb{R}^N\ : f(x) \leq \beta \} \end{equation} are compact. I need to ...
0
votes
1answer
64 views

How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
2
votes
1answer
76 views

Maximal Principle: Why using the new transition matrix $\tilde{P}$?

First some notation: Let $(X,E,P)$ denote a finite, irreducible Markov chain with finite state space $E$ and transition matrix $P$. Choose and fix a subset $E^°$ of $E$, which will be called ...
0
votes
4answers
326 views

prove there is no smallest positive rational number

How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
13
votes
6answers
2k views

Prove $1+2\sqrt3$ is not a rational number

How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.
0
votes
1answer
29 views

Best way to prove adjacency of a graph

Assume n is even. Considering a graph where each vertex in $v_1,...,v_n$ is adjacent to the next (ie $v_i \sim v_{i+1}$ for $1\leq i<n$) and where $v_1,v_n$ are each connected to at least $n/2$ ...
5
votes
5answers
136 views

Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is ...
1
vote
1answer
26 views

I have a problem understanding conceptually > using natural numbers

I am learning proofs with $\mathbb N $. I don't have significant problems using the axioms to prove propositions, I have a problem understanding certain axioms and the definition of >. 1) If $m,n ...
0
votes
1answer
42 views

I need to prove the following

I have the following proposition: For all $m,n \in\mathbb Z$, $-m < -n$ if and only if $m > n$. I know that if and only if means that I need to prove both ways, right? Proof: \begin{align*} -m ...
2
votes
0answers
25 views

Help on proving an observation related to pythagorean triangles.

Working for a while on Pythagorean triangles I observed that if $n$ has the prime factorization $p_1^{r_1}p_2^{r^2}...p_i^{r_i}$ where $r_k$'s not all even Then we have: ...
2
votes
5answers
137 views

Prove that that $x^2 = - 1$ has no solution in $\mathbb{Z}$

I have this proposition to prove: The equation $x^2 = -1$ has no solution in $\mathbb Z$. I was told that this is an opportunity for a proof by contradiction. I have already proven that for $m ...
0
votes
1answer
27 views

Proof: Finite product of elementary matrices.

I need to proof the following theorem: Let $A,B \in M_{mxn}(\mathbb{R})$ and $B$ is $A \sim B$ (equivalent of rows with $A$), then there exist a matrix $C \in M_{m}(\mathbb{R})$, such that $B = CA$, ...
0
votes
2answers
33 views

Let $f$ have a jump discontinuity at $x_0$. Show that $f(x_1), f(x_2), \ldots$ has at most two limit-points.

This is a question I understand intuitively but am having trouble proving rigorously: Let $f$ have a jump discontinuity at $x_0$. Show that if $x_1, x_2, . . .$ is any sequence of points in the ...
2
votes
2answers
48 views

Simple Division Proof

Prove that for every three integers i, j, and k, if i $\nmid$ jk, then i $\nmid$ j We've just started proofs and I am at a complete loss for how to go about doing it. I've tried proving through ...
0
votes
3answers
35 views

Surjective linear transformation in $\mathbb R^2$ is injective

Let $T:\mathbb R^2 \to \mathbb R^2$ be a surjective linear transformation. Prove that $T$ is injective without using the rank-nullity theorem Is there a way to prove this theoremn, without the ...
1
vote
1answer
31 views

Lemmas to characterize Eistenstein primes

Here are lemmas to characterize Gaussian primes Lemma1 Let $p$ be an odd prime such that $p\equiv 1 \pmod 4$. Then, the Legendre symbol $(-1/p)=1$ Lemma2 Let $p$ be an odd prime such that ...
3
votes
2answers
49 views

Prove that $ \cos x - \cos y = -2 \sin ( \frac{x-y}{2} ) \sin ( \frac{x+y}{2} ) $

Prove that $ \cos x - \cos y = -2 \sin \left( \frac{x-y}{2} \right) \sin \left( \frac{x+y}{2} \right) $ without knowing cos identity We don't know that $ \cos0 = 1 $ We don't know that $ \cos^2 x + ...
-2
votes
2answers
69 views

The smallest non-abelian group $G$ with a non-normal subgroup [closed]

This time I need to find the smallest non-abelian group $G$ with a non-normal subgroup, then my questions are: 1)Can someone help me to find it? 2)Once we find it, How Can you prove that it is the ...
2
votes
2answers
196 views

How to prove a subspace is non empty?

To prove that a subspace W is non empty we usually prove that the zero vector exists in the subspace. But then is it necessary to prove the existence of zero vector. Can't we prove the existence of ...
1
vote
2answers
101 views

Why can partial derivatives be exchanged?

In the Equality of mixed partial derivatives post in this stack exchange, one of the answers to the questions of do partial derivatives commute is: Second order partial derivatives commute if f is ...
0
votes
2answers
21 views

Method of proof confusing between Vector space and Linear Transformation.

I'm confusing about the way to determine the Vector space and the Linear Transformation. My knowledge is the way to determine whether the map given is Linear Transformation by proving this : ...
1
vote
3answers
42 views

Prove if $n=p_1p_2\cdots p_k +1$, then for every $i$, $i=1,2,\cdots k, p_i$ does not divide n.

I am trying to prove Let $p_1, p_2, \cdots p_k$ be prime integers. if $n = p_1p_2\cdots p_k + 1$ then for every $i$, $i =1, 2\cdots,k$, $p_i$ does not divide n. I start with contradiction; ...
2
votes
1answer
103 views

Fourier series: Understanding a proof

Let $f:[0,2\pi]\to\mathbb{R}$, continuous, such that for all $n\in\mathbb{Z}$:$$\int_0^{2\pi} f(x)e^{i(n+\frac{1}{2})x} dx = 0$$ Prove that $f(x)=0$. The solution: We can rewrite the integral ...
1
vote
1answer
28 views

prove that $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$ [duplicate]

How to prove that $f: \mathbb R^2 \to \mathbb R$ $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$? By definition: Let $\epsilon>0$ I need to prove ...
4
votes
1answer
82 views

Fundamental proof of Taylor's theorem using little-o notations

Is there a fundamental proof of Taylor's theorem using little-o notation? I assume $f:E\rightarrow F$ as a mapping between Banach spaces and write $(h^i)$ for $(h,\ldots,h)$ ($i$ times iterated). ...
1
vote
1answer
33 views

A question about $(\exists x: p(x) \supset q(x)) \equiv ((\forall x: p(x)) \supset (\exists x: q(x)))$

Conider the usual logical connectives $\wedge$ for "and", $\vee$ for "or", $\supset$ for material implication, $\equiv$ for material equivalence, $\neg$ for "not" et cetera. We all know and use laws ...
0
votes
2answers
38 views

Show that $f(z)=2z+z^2$ with $|z|<1$ is a one-to-one function

Show that $f(z)=2z+z^2$ with $|z|<1$ is a one-to-one function. By using $z=x+iy$, I get $$f(z)=x^2+2x-y^2+i(2y+2xy)$$ So to prove that this function is one-to-one, suppose $f(z_1)=f(z_2)$ where ...
4
votes
6answers
190 views

Prove $1+\sqrt2$ is irrational

I am trying to prove $1+\sqrt 2$ is an irrational number. I start with contradiction Proof: assume that $1+\sqrt 2$ is a rational number such that $1+\sqrt 2=\frac{m}{n}$ where m and n are some ...
2
votes
2answers
48 views

Proving square of nonzero integer is natural number

I am learning proofs with $\mathbb N$ and have this proposition: Let $m \in\mathbb Z$. If $m \ne 0$, then $m^2 \in\mathbb N$. Previously, I have proven: For $m \in\mathbb Z$, one and only one of the ...
1
vote
2answers
55 views

Prove $f$, satisfying $\left|f(x)-f(y)\right|\le K\left|x-y\right|^{\alpha}$, is constant. Proof strategy.

Let $\alpha>1, K>0$ and let $f:[0,1]\to \Bbb{R}$ satisfy $$\left|f(x)-f(y)\right|\le K\left|x-y\right|^{\alpha}, \forall x,y\in [0,1].$$ Prove $f$ is constant. What I basically need is a proof ...
1
vote
2answers
38 views

Prove that $B = \bigcup\{A_\alpha \mid \alpha \in[1,2]\}$

I am working this question: Set $B = \{(x, y)\mid 1 \le x^2 + y^2 \le 4\}$, $A_\alpha = \{(x, y)\mid x^2 + y^2 = α^2\}$. Prove that $\bigcup\{A_\alpha\mid \alpha \in [1, 2]\} = B$. because this ...
7
votes
0answers
224 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
5
votes
1answer
73 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
3
votes
2answers
70 views

Proving $2^n > (n+1)^2$ for $n\geq 6$ by induction.

So here is what I have to prove by induction: $2^n\gt(n+1)^2$ for $n\ge6$ So, first lets say $n=6$ $$2^6\gt(6+1)^2$$ $$64\ge49$$ Now, assume $n=k$ $$2^k\gt(k+1)^2\text{ for } k\ge6$$ Prove ...
0
votes
0answers
32 views

Does ℘(A ∩ B)=℘(A) ∩ ℘(B) hold? How to prove it? [duplicate]

I'm currently working on some discrete mathematics work and I've encountered a question I'm not sure how to answer exactly. Precisely, I'm trying to prove that two power, intersected sets statements ...
1
vote
3answers
85 views

Is this a legal way to prove an inequality?

I have to prove the following inequality: $(x+y)\sqrt{\frac{x+y}{2}}\geq x\sqrt{y}+y\sqrt{x}$ where $x,y>0$. After squaring both sides I obtain: $(x^2+2xy+y^2)\frac{(x+y)}{2}\geq x^2y+xy^2$ ...
0
votes
1answer
122 views

Existence of the centre of mass of a measure

Let $\mu$ be a finite Borel measure on $\mathbb R^n$ with compact support. Show that there exists $b \in \mathbb R^n$, the centre of mass of $\mu$, such that $$b \cdot v = \frac{\int x \cdot ...
2
votes
1answer
31 views

why conventional approximation method is true?

why the text book method for finding the fitting curve is right ? we have n data we want to approximate with a polynomial of degree m $P_m(x)$ (m < n-1). and of course $E = \sum_{i=1}^m ...
0
votes
1answer
20 views

Proving an Inequality

Suppose I've got a positive number $a$>0. I've got some other positive number $b$>0. I am given that $a$ > $b$. How do I prove that the above implies $-a + \sqrt{ a^{2} - b^{2} }$ < 0 ? So far I ...
0
votes
0answers
60 views

squeeze theorem - math

I am trying to prove the following: 1/ek <= (1/k)(1-(1/k))^(k-1) <= 1/2k for k>=2 in doing so I tried induction proof, and contradiction and it didn't work, it gets too complicated... Then ...
1
vote
1answer
61 views

Establish the convergence or divergence of a sequence [duplicate]

Establish the convergence or divergence of the sequence (y_n), where: y_n := 1/(n+1) + 1/(n+2) + ... = 1/(2n) for n /in N.
2
votes
2answers
77 views

Proving $a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$ without Euler's Theorem

Is there a different way of solving $$a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$$ without the use of Euler's Theorem (or proof of Euler's theorem for $p^k$)? I've tried to use the Chinese Remainder ...
0
votes
0answers
41 views

Corollary of the inverse function theorem

Let $U\subset \mathbb{R}^{n}$ and $ f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...