For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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4
votes
2answers
125 views

Help to find all different cases need for proof about homomorphism from Z to R

I am a bit confused about why my professor approached the following a certain way, and also why it cannot be done differently. The question is to prove that for any ring R we there is a unique ...
4
votes
1answer
97 views

Prove a piecewise function is integrable

Let the function: $$ f(x) = \begin{cases} 1 &\mbox{if } x = \tfrac{1}{n}, n\in\mathbb{N} \\ 0 & \text{otherwise} \end{cases} $$ Show that $f(x)$ integrable and evaluate $$\int_0^1 f(x)...
0
votes
0answers
31 views

Upper Riemann integral equals upper Riemann Sums

In Analysis 1 by Terrence Tao the definition of the Upper Riemann integral, is given as follows Let $f : I \to \mathbb{R}$ be a bounded function on a bounded interval $I$. We define the Upper ...
0
votes
0answers
16 views

Looking for errors in my proof that the normal vector to a path is zero.

Let $\mathbf c : \mathbb R \rightarrow \mathbb R^n$ be a smooth ($C^\infty$) path where $\mathbf c(t) = \left(c_1(t), c_2(t), \ldots, c_n(t) \right)$. Let $\mathbf T(t)$ be defined as the normalized ...
0
votes
1answer
40 views

Real-valued, differentiable function with bounded derivative is uniformly continuous

Real-valued, differentiable function on $\mathbb{R}$ with the bounded derivative is uniformly continuous. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function such that there is $M&...
1
vote
0answers
77 views

My proof of Birkhoff–von Neumann theorem whit a probabilistic point of view

I have just found a very beautiful and short proof for the birkhoff-von Neuman theorem that gives a new probabilistic approach. Notations : Let, $S_{n}$ be the set of permutations of the set {$1,...,...
2
votes
3answers
114 views

Show $f(x)=\sqrt{x}$ is continuous on $[0,1]$

Show $f(x)=\sqrt{x}$ is continuous on $[0,1]$ For this question, I tried two ways to do it. Suppose that sequence $\{x_n\}$ converges to $x_0\in[0,1]$ and $x_n\in[0,1]$ for all $n$. Then we ...
2
votes
1answer
79 views

Proving the inverse of a bijection is bijective

Let $f: A\to B$ and that $f$ is a bijection. Show that the inverse of $f$ is bijective. Surjectivity: Since $f^{-1} : B\to A$, I need to show that $\operatorname{range}(f^{-1})=A$. But since $f^{-1}$...
1
vote
0answers
23 views

Projection of a point onto a sphere proof verification?

I wanted to work out how to project plane curves onto a sphere, and I started with projecting a single arbitrary point uniquely onto a sphere, and I arrived at the following conclusions. Can someone ...
0
votes
1answer
32 views

if $x_0$ is a limit point of $D$, then a function $f:D\rightarrow\mathbb{R}$ is continuous at $x_0$ iff $\lim\limits_{x\rightarrow x_0}f(x)=f(x_0)$

Prove that if the point $x_0$ in $D$ is a limit point, then a function $f:D\rightarrow\mathbb{R}$ is continuous at $x_0$ iff $\lim\limits_{x\rightarrow x_0}f(x)=f(x_0)$ $\Longrightarrow$ Since the ...
0
votes
2answers
21 views

Equivalence relations and classes

Let $S = [10]$ (the set containing the integers 1,...,10). Define an equivalence relation R on S by $xRy$ $iff$ $x^2 \equiv y^2$ (mod 5) is an equivalence relation and determine the equivalence ...
1
vote
0answers
35 views

Prove that$f(a)<f(c)<f(b)$ where the function $f[a,b]\rightarrow\mathbb{R}$ be continuous and one to one and such that $f(a)<f(b)$.

Let the function $f[a,b]\rightarrow\mathbb{R}$ be continuous and one to one and such that $f(a)<f(b)$. Let $c$ be a point in the open interval $(a,b)$. Prove that$f(a)<f(c)<f(b)$. Suppose ...
3
votes
4answers
301 views

Proving a summation inequality with induction

The exact question: Prove: $\displaystyle\sum_{k=1}^n \frac{1}{\sqrt{k}}\gt2(\sqrt{n+1}-1)$ I have looked at similar problems but still don't understand how to prove this inequality by induction. ...
0
votes
0answers
48 views

Definition of renorming of a space

The following is Proposition $4.5$ from Kalton's paper. Let $X$ and $Y$ be Banach spaces such that there exists a Lipschitz embedding $L:X \rightarrow Y$ such that $L(0)=0$ and $\overline{span(...
5
votes
1answer
110 views

Munkres exercise

Problem Let $\{A_\alpha\}$ be a collection of subsets of $X$; let $X=\bigcup_{\alpha}A_\alpha$. Let $f:X\rightarrow Y$;suppose that $f\vert_{A_\alpha}$ is continuous for each $\alpha$. An index ...
2
votes
2answers
120 views

Is the argument used by the Numberphile video to show that $1 + 2 + 3 + 4 + \dots = -1/12$ valid? [duplicate]

At the end of the Wikipedia article on: $$1+2+3+4 +\dots$$ an argument is present that the sum adds up to $-\frac{1}{12}$. Here is the numberphile video. Here's my attempt to fill in the ...
0
votes
1answer
52 views

Prove there is a unique function $g : B \rightarrow A_1 \times \dotsc \times A_n$

I was wondering if someone would not mind proofreading my demonstration for the following problem. Any sentences in brackets [] will be omitted in the formal proof. Problem Let $B$ be a set, let $...
1
vote
1answer
64 views

A group acts transitively on a finite set

Question: Assume the group G acts transitively on the finite set A and let $H\unlhd G$. If $\{\Theta_1 , ... , \Theta_n\}$ are the distinct orbits of H on A, prove that G acts transitively on these ...
0
votes
1answer
32 views

Proof that composition on $Aut(G)$ is associative

Would it be correct to prove it the way below? I'm feeling somewhat ambiguous as to the correctness of my proof. I would appreciate your input. Proof: We need to prove that for $\phi$, $\psi$, $\rho ...
0
votes
2answers
29 views

Set Theory: Distributive laws with respect to the subtraction of sets

"Prove that $A \cap (B - C) = (A \cap B) - (A \cap C)$" Now what I do is the following, and what I get is a bit different. We know that for some arbitrary member $x$, $x \in A$ and simultaneously $x \...
1
vote
0answers
33 views

Is this proof that the subsequences of a convergent sequence converge to the same limit rigorous?

I am working through Abbot's Understanding Analysis and I am confused about when a proof is rigorous enough to be acceptable. This is a proof I came up with to show that subsequences of convergent ...
1
vote
2answers
112 views

Two different results of calculation of the limit of $(2+4+\dots+2n)/n^2$

I would like to know where is the problem when i calculated those two limit with the following ways : method (1): $$\lim_{n\to \infty} \dfrac { 2+4+6+\cdots 2n} {n²} = \lim_{n\to \infty} \dfrac {n(n+...
1
vote
1answer
85 views

Show $h(x)=1/(x^2+1)$ is uniformly continuous on $\mathbb R$.

Show $h(x)=1/(x^2+1)$ satisfies the $\epsilon-\delta$ definition for uniform continuity on $\mathbb{R}$ Attampt: Let $\epsilon>0$, there exists a $\delta=\epsilon/2$ such that $|x-y|<\...
0
votes
1answer
23 views

Is this the correct way of finding the inverse of a cycle?

If I have a cycle $$g=(132)$$then the inverse cycle is found by writing it in a backwards order, in other words $$g^{-1}=(231)$$ Say now we have $g=(135)(24)$ and I want to find what $g^{-1}$ is. ...
0
votes
0answers
32 views

Palindrome Proof,general complete induction, forward&backwards

$\sum$ is an Alphabet. $\sum^*$ is the set of strings formed from symbols in $\sum$ The set of palindromes of length n, $P^n$ is inductively defined as: $P^0$ :={\varepsilon} $P^1 := \left\{a\...
0
votes
0answers
55 views

$f([a,b])=[f(a),f(b)]$ for continuous strictly monotonic function

Let $f$ is continuous and strictly monotonic function on $[a,b]$. Prove that $$f([a,b])=[f(a),f(b)].$$ Proof: We know that $[a,b]$ is a connected set in $\mathbb{R}$ then $f([a,b])$ is connected ...
3
votes
1answer
66 views

All functions such that $f'(x) = f(x+1)-f(x) = \frac{f(x+2)-f(x)}{2}$ for all $x \in \mathbb{R}$

I would like to find all (differentiable) functions $\mathbb{R} \to \mathbb{R}$ satisfying $$f'(x) = f(x+1)-f(x) = \frac{f(x+2)-f(x)}{2}$$ for all $x \in \mathbb{R}$. I claim that the only functions ...
3
votes
1answer
98 views

A group homomorphism $\phi : G\to H$. Then $\phi[G]$ is abelian iff $xyx^{-1}y^{-1}\in \ker(\phi)$.

From A First Course In Abstract Algebra by John B. Fraleigh, 7th edition. Section 13, problem 50. Let $\phi: G\to H$ be a group homomorphism. Show that $\phi[G]$ is abelian if and only if for all $x,...
0
votes
1answer
44 views

Understand a proof that if $X$ is a Banach space, and $E$ is a closed subspace of $X$, then the quotient space $X/E$ is also a Banach space.

Trying to understand a proof that if $X$ is a Banach space, and $E$ is a closed subspace of $X$, then the quotient space $X/E$ is also a Banach space. Now they let $\{Y_n\}$ be a Cauchy sequence in $...
0
votes
1answer
39 views

Gaussian Primes of the form $a^2 + 1$ where $a \in \mathbb{Z}$

Are there any primes of this sort? I know that a prime $p \in \mathbb{Z}$ is composite in $G$ iff $p$ is a sum of $2$ squares. Thus, as $a \in \mathbb{Z}$, $(a^2 + 1) \in \mathbb{Z}$, by letting $p = (...
0
votes
2answers
39 views

4 Divides x Proofs of conjectures

Hi there I'm working on a set of problems and I'm having some difficulty proving and disproving these examples. I know that #1 is essentially (There exists K where [x=4k]) I'm lost after that. I'm not ...
1
vote
1answer
59 views

How to show uniform convergence of this series of functions

I was working the following question from a previous qualifying exam, the solution below seems to have a minor snag in it below Let $f_{1}: [a,b] \rightarrow \mathbb{R}$ be Riemann integrable ...
1
vote
1answer
31 views

How to prove that $f: \mathbb{Z} \to \mathbb{N}$, $f(x)=x^x$ is one to one and not onto

I tried to solve it by first showing that the function is monotonically increasing which shows that it is one to one. Then finding a number that lies between two consecutive values of the function. 1)...
1
vote
1answer
47 views

Positive function with finite integral is finite almost surely

Let $\displaystyle f:(X,\Sigma, \mu)\to (\bar{\mathbb R_+}, \mathcal B)$ be a measurable non-negative function that may assume the value $+\infty$. Suppose that $\int f d\mu$ is finite. Prove ...
2
votes
1answer
42 views

If $R$ is an integral domain, then $(R[x])^\times=R^\times$

If $R$ is an integral domain, then $(R[x])^\times=R^\times$ So since $R$ is an integral domain, it follows that $R[x]$ is an integral domain. We have $f(x)g(x)=1$ then we know that $\deg(f(x)g(x))= ...
0
votes
1answer
51 views

Showing $U(20)$ is not cyclic: Is my proof correct?

I solved the following exercise: Exercise: Show that $U(20) \neq \langle k \rangle$ for any $k \in U(20)$. Here $\langle k \rangle $ is defined to be the set $\{k^i \mid i \in \mathbb Z \}$. My ...
0
votes
0answers
47 views

$\sum \zeta (s)$ converging proof?

I was pondering about the fact that the Zeta function can be represented as an infinite sum, $\zeta (s) = 1/1^s + 1/2^s + 1/3^s +\ ...$, and I thought about the infinite sum of the zeta function ...
0
votes
2answers
29 views

Please could someone check my answers to these basic group theory exercise

So I've tried to solve this exercise and I was wondering if someone would check my answer and tell me if it is correct please? Exercise: Part 1: Suppose $a$ is a group element such that $a^6 = e$....
2
votes
1answer
32 views

Proof that disjoint cycles commute

I'm trying to prove that in a symmetric group two disjoint cycles commute. But I suspect that something is not right about my proof (a sense of vagueness). Some hints would be appreciated. Here's my ...
6
votes
1answer
205 views

Extension of a linear map

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $Lip_0(X)$. The set Lip$_0(X)$ is the set of all real-valued ...
0
votes
2answers
74 views

Show that for any natural number $n>24$ there exist natural numbers $p$ and $q$ such that $ n=5p+7q$

Show that for any natural number n>24 we have : $n=5p+7q$ such that $p$ and $q$ are natural. I tried using induction 1) for $n=24$ we have $n=(7 \cdot 2)+(5 \cdot 2)$ 2) we suppose that $n=5p+7q$...
1
vote
3answers
39 views

How do I complete a proof for an inequality by using the triangle inequality theorem?

Prove that the inequality will hold for every real number, $x$ $$\left| 2+x \right| \le \left| 2x+1 \right| +\left| 1-x \right| $$ Proof: This proof is by case analysis. Case 1: 1) Let $a=2x+1$ ...
0
votes
1answer
196 views

Hollow matrix and n-th power proof

I have this exercise but I don't have acquired yet all the information I need in order to deal with it, at least. Show that if an upper (lower) triangular matrix has only zeros on the main diagonal, ...
0
votes
1answer
43 views

Proof that a certain bounded set has no maximum seems wrong

if $A = (0, 1)$ then $1$ is the supremum, but there is no maximum! To show this, assume for a contradiction that $x \in A$ is the maximum of $A$. Then also $$(1+x)/2 \in A \quad \quad (1)$$ and $$(1+...
3
votes
1answer
177 views

Can be justified this $\zeta(3)=\int_0^1\frac{1}{x}\sum_{n=0}^\infty\frac{x^{(n+1)^{3/2}}}{(n+1)^{3/2}}dx$?

It is well known the unsolved problem concerning to Àpery constant $\zeta(3)=\sum_{n=1}^{\infty}\frac{1}{n^3}$ (see this site or [1] for example). The following computations will be viewed with the ...
6
votes
1answer
134 views

Show the equivalence of these definitions of independence of random variables

Let $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. There are two different definitions of the independence of random variables $X_1, X_2, ...$ on $(\Omega, \mathscr{F}, \mathbb{P})$: ...
0
votes
3answers
68 views

Understanding of a proof about upper/lower triangular matrices property under multiplication

Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular. I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it ...
2
votes
2answers
186 views

Is there a mistake in this proof?

Proposition 1: The number of occurrences of an event within a unit of time has a Poisson distribution with parameter $\lambda $ if and only if the time elapsed between two successive occurrences ...
1
vote
2answers
71 views

Show that if $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous, then sequence $f(x+1),f(x+\frac{1}{2}),f(x+\frac{1}{3}),\ldots$

Show that if $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous, then sequence $f(x+1),f(x+\frac{1}{2}),f(x+\frac{1}{3}),\ldots$ is uniformly convergent. Let $g_n(x)=f(x+\frac{1}{n})$. A ...