0
votes
1answer
80 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
2
votes
3answers
72 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
1
vote
1answer
40 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
2
votes
1answer
73 views

Prove that the dual graph of any (planar) graph is connected

I'd like to know if there's a standard proof that the dual graph of any planar graph is connected (or, if there's a counterexample, I'd like to know that too). I've thought of a proof that might work ...
1
vote
0answers
80 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
0
votes
1answer
58 views

Prove that if $S$ has a greatest element $b$, then $b = lubS$.

Prove that if $S$ has a greatest element $b$, then $b = lubS$. These are the definitions I used: Def. Given a partially ordered set ($P, \le$), then an element $b$ of a subset $S$ of $P$ is the ...
0
votes
0answers
40 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
6
votes
0answers
73 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
12
votes
3answers
1k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
6
votes
3answers
235 views

Proving $fg$ and $f+g$ is Riemann integrable through the easy and hard way.

Problem: Suppose $f,g$ are Riemann integrable functions, show that $f+g$ and $fg$ are also Riemann integrable. I know there is really easy to do this with measure theory, but I want to see if ...
2
votes
2answers
109 views

Solve $ax \equiv b \mod m$ without Linear Congruence Theorem or Euclid's Algorithm?

Origin page 5. The overhead doesn't look like Linear Congruence Theorem or anything from Euclid's Algorithm. page 4 tries to delineate ...
0
votes
3answers
59 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
3
votes
1answer
39 views

A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
1
vote
0answers
27 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
0
votes
1answer
61 views

Inequality proof, why isn't squaring by both sides permissible?

Suppose $a$ and $b$ are real numbers. Prove that if $0 < a < b$ then $a^2 < b^2$. I understand that the normal way to prove this is to multiply $a < b$ by $a$ and then by $b$ and then ...
0
votes
1answer
32 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
2
votes
1answer
108 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
0
votes
0answers
38 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
1
vote
1answer
34 views

Proof that 6 divides $a \in \mathbb{Z}, a(a^2 - 7)$

I am trying to prove a question from my tutorial sheet, is this an acceptable proof? Six cases exist: $$a,k \in \mathbb{Z}, a(a^2 - 7) = 6k \\\text{Proof:}\\ a = 0 \mod 6 \longrightarrow a^2 = 0 \mod ...
2
votes
2answers
53 views

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$ My Proof: Ten cases exist, yielding the following equalities: $$(1\mod{10})^2 = 1\mod{10}$$ $$(2\mod{10})^2 = 4\mod{10}$$ ...
0
votes
2answers
49 views

Proof of the 'second' triangle inequality

I am trying to prove the 'second' triangle inequality: $$||x|-|y|| \leq |x-y|$$ My attempt: $$----------------$$ Proof: $|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = ...
2
votes
0answers
84 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
1
vote
1answer
61 views

Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
1
vote
0answers
35 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
0
votes
0answers
53 views

The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
2
votes
0answers
82 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
1
vote
1answer
157 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
1
vote
0answers
61 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
6
votes
1answer
279 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...
2
votes
1answer
51 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
1
vote
1answer
57 views

Prove that $ \lim (s_n t_n) =0$ given $\vert t_n \vert \leq M $ and $ \lim (s_n) = 0$

Let $ (t_n) $ be a bounded sequence, i.e., there exists $ M $ such that $ \vert t_n \vert \leq M $ for all $ n $, and let $ (s_n) $ be a sequence such that $ \lim s_n = 0 $. Prove $ \lim (s_n t_n) ...
2
votes
4answers
150 views

Prove that $ \lim_{n \rightarrow \infty } \frac{n+6}{n^2-6} = 0 $.

My attempt: We prove that $ \lim\limits_{n \rightarrow \infty } \dfrac{n+6}{n^2-6} =0$. It is sufficient to show that for any $ \epsilon \in\textbf{R}^+ $, there exists an $ K \in \textbf{R}$ such ...
4
votes
4answers
192 views

Prove that $ \displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $.

My attempt: We prove that $$\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $$ It is sufficient to show that for an arbitrary real number ...
4
votes
3answers
212 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational zeros theorem, we can conclude that $ \pm 1$ and $ \pm 23 $ are the only possible rational solutions*. ...
3
votes
1answer
98 views

Proving that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1$ through an alternative method?

I'm reading the Berkeley Problems in Mathematics book: Prove that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1.$ I could find other ways to prove it, ...
2
votes
2answers
72 views

Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
1
vote
2answers
308 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
9
votes
1answer
160 views

Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$

I have to show that $$\sqrt[7]{7!} < \sqrt[8]{8!}$$ and I did the following steps \begin{align} \sqrt[7]{7!} &< \sqrt[8]{8!} \\ (7!)^{(1/7)} &< (8!)^{(1/8)} \\ (7!)^{(1/7)} - ...
2
votes
2answers
318 views

Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
2
votes
0answers
127 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
0
votes
0answers
89 views

Proving that morphism of sheaves is iso iff induced morphism on stalks is iso

Is the following proof sound/does anyone have another more elegant (categorical) proof? The direction $\Rightarrow$ is obvious the "family of stalks"-functor is a functor and functors take isos to ...
3
votes
1answer
130 views

Proofs regarding Continuous functions 2

I need verification for this proof: Q: Suppose $f: (0,1)\rightarrow \mathbb{R}$ is defined by $f(x) = \begin{cases}\frac{1}{n} & \text{if }\text{x is rational with x} = \frac{m}{n}\text{ in ...
1
vote
1answer
79 views

Proofs regarding Continuous functions 1

Q: Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a bounded function (that is, there exists some $M\geq 0$ so that $|f(x)|\leq M$ for all $x\in\mathbb{R}$). Define a new function ...
0
votes
1answer
95 views

Alternatinve proof for the principle of the Iterated Suprema

The back of the book gave a proof similar to the proof here Proving principle of the Iterated Suprema, but I proved it following way before I checked the back of the book. Could some one verify this ...
0
votes
1answer
68 views

Examples of sets which measure cannot be obtained by discretisation

I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set $E\subset$ ...
0
votes
1answer
108 views

Showing that $a_n \not \to 17$ implies a subsequence $a_{n_k}$ that is $\epsilon$ far from $17$ for some $\epsilon > 0$

I want to check my proof for this question: Suppose a sequence {$a_n$} does not converge to 17. Prove that there exists some $\epsilon$ > 0 and a subsequence {$a_{nk}$} so that $|a_{nk} - 17|$ > ...
2
votes
2answers
201 views

Verification of proof of the Sequence of Arithmetic Theorem

Suppose $\left\{b_{n}\right\}$ is a sequence of real numbers which converges to $M$, so that $b_{n} \neq 0$ for each $n$, and $M \neq 0$. Prove that the sequence $\{ \frac{1}{b_n} \}$ converges to ...
2
votes
1answer
236 views

Proof for $-\sup(A) = \inf(-A)$

Let $A$ and $-A = \{ -x \mid x \in A \}$ be two bounded sets. I have to prove that $-\sup(A) = \inf(-A)$, i did it in the following way and wish to know if it is sufficient: $ \exists x\in A$ such ...
5
votes
1answer
206 views

Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs

$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}$ $\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
2
votes
1answer
117 views

Every finite group is finitely generated. - alternate proof

Is that's all? Thank you. :-) A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely ...