0
votes
2answers
28 views

Convergence of integral, that is absolutely convergent, proof

Can you think of any proof on convergence of improper integral, that is absolutely convergent? It is so obvious, that I really don't know where to start. Triangle inequality gives us ...
4
votes
0answers
33 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
1
vote
2answers
62 views

Exercise about truth functions in J.R.Shoenfield's “mathematical logic”

The first exercise in Joseph R. Shoenfield's "mathematical logic" is: An n-ary truth function $H$ is definable in terms of the truth functions $H_1,\dots,H_k$ if $H$ has a definition ...
1
vote
2answers
45 views

Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$. The proof is easy enough ...
3
votes
2answers
39 views

Prove that there is no bipartite graph on $14$ vertices with this degree sequence.

Prove that there is no bipartite graph on $14$ vertices with degree sequence: $$6, 6, 6, 6, 6, 6, 6, 6, 5, 3, 3, 3, 3, 3.$$ I assume a vertex with degree $5$ breaks this graph from being ...
0
votes
2answers
38 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
0
votes
1answer
55 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
6
votes
1answer
74 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
1
vote
3answers
40 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
1
vote
2answers
48 views

Discrete Mathmatics Proof

Here is the question: $a$ and $b$ are any two integers. $c$ is any prime. Prove that if $c$ divides $ab$, then $c$ divides $a$ or $c$ divides $b$ (or both, as in it can divide either or both, i.e. ...
0
votes
2answers
24 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
2
votes
1answer
54 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
1
vote
1answer
38 views

Proving the Well-Ordering Property

My textbook states the Well-Ordering property as following: If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \le a$, for all $a$ in $A$ ($m$ is called ...
1
vote
0answers
59 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
2
votes
2answers
39 views

Proof About Division of Integers

Here is a problem I just finished working on: Prove that if $n$ is composite then there are integers $a$ and $b$ such that $n$ divides $ab$ but not $n$ does not divide either $a$ or $b$. One ...
0
votes
1answer
21 views

Integrability condition on the Fourier transform

I have a question for This is much healthier who answer my query- This is a problem from the book from Stephane Mallat "A wavelet Tour of signal processing: a sparse way". A function $f$ is bounded ...
0
votes
1answer
57 views

My proof is wrong, can anyone tell me why?

$$\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}, [x(x+1) = y(y+1)] \Leftrightarrow [x = y]$$ $$\forall x \in \mathbb{Z} , \forall y \in \mathbb{Z}, [x(x+1)=y(y+1)]\Leftrightarrow [x=y]$$ ...
0
votes
1answer
43 views

Trigonometry - proving an inequality

I came across this question while doing trigonometry. I have tried everything that I could possibly think of, AM/GM, converting it into quadratic equation, conditional identities, solving from RHS, ...
1
vote
1answer
24 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
1
vote
1answer
39 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
0
votes
2answers
48 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
1
vote
3answers
53 views

Proof for modulus via direct or contrapositive

I have to prove the following via direct proof or via contra positive. For $a,b\in \mathbb{Z} $ it follows that $ (a+b)^3 \equiv a^3 + b^3 \mod 3$ I'm unsure of the best way to approach this ...
1
vote
1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
1
vote
3answers
30 views

Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
1
vote
1answer
31 views

Correctness of Proof by Refutation

I am trying to solve the following by proof by refutation: A or B 1. NOT A or I 2. NOT B or T 3. NOT I 4. NOT T 5. Where the goal is to prove a contradiction. ...
4
votes
1answer
160 views

How to prove that equilateral triangle formed by cube's corners cannot be fully inserted to this cube

I would like to prove that equilateral triangle prescribed by cube's corners and sides equal to $b = a \sqrt{2}$ cannot be inserted into the interior of a cube of side $a$. This triangle is presented ...
1
vote
0answers
54 views

Is the proof of non-totality of a function $f$ trivial if $f$ is recursively defined only on a subset of $\Bbb N$?

I define a fuction $*$ in a recursive way on a subset of natural numbers (with subset I mean that is $*$ define in the second argument only for $\Bbb N-\{0,1\}$ in other words we have $*:\Bbb N ...
2
votes
1answer
80 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 ...
2
votes
0answers
39 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
1
vote
3answers
29 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
3answers
47 views

Disproving 0 as a dividend

Prove each of the following statements. (a) For all $b \in \mathbb{Z}$ if for all $k \in \mathbb{N}$, $b \not\mid k$, then $b = 0$. By hypothesis: $b \not\mid k \implies b\ell \neq k, \ell ...
2
votes
1answer
68 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
0
votes
1answer
70 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
vote
1answer
14 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
votes
1answer
21 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
1
vote
1answer
52 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
0
votes
1answer
32 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
0
votes
1answer
30 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
2
votes
2answers
294 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
1
vote
2answers
23 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
2
votes
2answers
22 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
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 $|y|\ge y$, ...
0
votes
1answer
35 views

Are these proofs correct?

I haven't formally learn how to do proofs, but I attempted some of these. It'd be great if you guys can check them and give me some pointers. Thanks!
1
vote
2answers
64 views

Ground Plan - Prove Fermat-Euclid's Totient Theorem with Lagrange's Theorem

If $\gcd(a,n) = 1$, then $a^{\phi(n)}\equiv 1\pmod n$. Here's a three-step proof. An integer a is invertible means there's some $a^{-1}$ such that $aa^{-1}\equiv 1 \pmod n$. By cause of Jones p84 ...
4
votes
2answers
168 views

Fermat's Little Theorem fails for composite instead of prime numbers.

I know Fermat's Little Theorem = Fermat-Euler's Totient Theorem when $n$ is prime. Elementary Number Theory, Jones, p83 writes if we simply replace p with a composite integer n, then the ...
2
votes
1answer
67 views

Backward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

(1) How can you preconceive to prove by contradiction? Prove by contradiction. Suppose $n$ is composite. This means there exists a divisor $d|n$ such that $1<d<n$. We are given that ...
1
vote
2answers
41 views

Ground Plan – Forward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

Lemma 5.3 - I omit proof here - Let p be prime. Then $x^2 \equiv 1 \, (mod p) \iff x \equiv \pm 1 \; (mod p)$ First we establish the result for the first two primes 2, 3. Then prove the result for ...
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) ...