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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1
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1answer
31 views

Proving that $O(n)$ is compact

Let $O(n)$ denote the group of orthogonal matrices under multiplication. We want to show that this is set is compact. To show $O(n)$ is compact, we can use Heine-Borel and show that it is closed and ...
6
votes
3answers
203 views

Conjectured compositeness tests for $N=k \cdot 2^n \pm c$

How to prove that these conjectures are true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ...
3
votes
4answers
100 views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part ...
2
votes
1answer
47 views

Let p<q both be prime numbers. Prove that log is not rational number

So i was given a question that starts off like this Prove that $\log_q(p)$ is not a rational number. Recall that $\log_y(x)$ for real numbers $x,y>0$ is defined to be the real number $r$ so ...
0
votes
1answer
27 views

Computing the gcd of a relatively prime polynomial

I was given a question that starts off like this. Suppose that $a, b \in \mathbb{N}$ and relatively prime. For each of the following, if the answer must be one particular number, then compute it; ...
0
votes
1answer
27 views

Proving the set of finite subsets of $\mathbb{N}$ is countably infinite [duplicate]

So I was given a question that begins like this. Let $P_{\text{fin}}(\mathbb{N})$ be the following set (called the finite power set of $\mathbb{N}$): $$ P_{\text{fin}}(\mathbb{N}) = \{X ...
0
votes
1answer
17 views

Determining cardinality and inverse

Let the function $\chi: P(Z) \to P(Z)$ be defined by $\chi(B) = B^c$ for any $B \in P(Z)$. (In other words, $\chi$ sends a subset $ B \subseteq Z$ to its complement, $B^c$, i.e. the set $Z - B$.) ...
0
votes
2answers
39 views

Uniform Convergence and limit $(n+1)\int_0^1 x^nf(x) \; dx$ [duplicate]

If $f$ is a continuous real-valued function, show that $$ f(1)=\lim_{n\to \infty} \int_0^1 (n+1)\,x^n \,f(x) \; dx $$ I am looking for a general hint or steps to proceed but I want to fill them in. ...
0
votes
2answers
28 views

Proving a number can be chosen to a multiply a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function so that it never exceeds another function

Let $p,q,r$ be real numbers with $p,r,pr-q^2 > 0$. I am trying to prove $\exists\gamma> 0$ s.t. $\forall(x,y) \in \mathbb{R}^2 . px^2 + 2qxy + ry^2 \geq \gamma(x^2+y^2).$
9
votes
6answers
164 views

Follow-up Question: Proof of Irrationality of $\sqrt{3}$

As a follow-up to this question, I noticed that the proof used the fact that $p$ and $q$ were "even". Clearly, when replacing factors of $2$ with factors of $3$ everything does not simply come down to ...
2
votes
2answers
60 views

Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$

This is Velleman's exercise 3.4.13: Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$. I am am stuck on that one. Seems like I am ...
1
vote
2answers
78 views

Explain the proof of irrationality of $\sqrt{2}$

How does this proof show the irrationality of $\sqrt{2}$ ? I am new to proofs and don't really understand the logic used here.
3
votes
7answers
118 views

Proving $\frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \cdots + \frac{1}{n\cdot(n+2)} = \frac{3}{4} - \frac{(2n+3)}{2(n+1)(n+2)}$ by induction for $n\geq 1$

I'm having an issue solving this problem using induction. If possible, could someone add in a very brief explanation of how they did it so it's easier for me to understand? $$\frac{1}{1\cdot3} + ...
4
votes
2answers
49 views

Find all $n \in \mathbb{Z}_{>0}$ such that $n^2+a \mid n^3+a$

Could anyone advise me how to find all $n \in \mathbb{Z}_{>0}$ such that $n^2+a $ divides $ n^3+a,$ where $a \in \mathbb{Z} \setminus \{0\}$ is fixed ? I have checked it is necessary that $n^2+a $ ...
0
votes
0answers
37 views

minimum number of leaves in a perfect binary tree

I'm trying to prove that the number of leaves in a perfect binary tree is at least H+1 where H is the height of the tree. This is what I've done up til now: No of leaves at height $H = 2^H$ Base ...
0
votes
0answers
23 views

Extending Monomorphisms from a Field to an Algebraic Closure

Let $F$ be a field, and $E/F$ a separable extension with degree $n$. If $\sigma$ is an embedding of $F$ into its algebraic closure $L$, then $\sigma$ extends to exactly $n$ embeddings from $E$ to ...
2
votes
2answers
91 views

Proof by contradiction problem

How can I prove that the statement: 'there is no integer, $n$, such that $4n^2 + 1 < 4n$' is true, by contradiction? I greatly appreciate any help with this. Thanks.
2
votes
1answer
53 views

Why does this two-element field not have a supremum if we disregard the Completeness Axiom for the reals?

I have two questions, and I address them in two seperate paragraphs below. I read that if we do not accept the Completeness Axiom for the reals and fields but accept all the other axioms for fields ...
1
vote
0answers
56 views

Related Integrals

I recently came across this fascinating problem online: Let $\displaystyle g_1(x) = \int_{0}^{x} g(t)\, dt,\: \: g_2(x) = \int_{0}^{x} g_1(t)\, dt,..., \: \: g_n(x) = \int_{0}^{x} g_{n-1}(t)\, dt$ ...
0
votes
0answers
30 views

Equivalence of Norms and Open Mapping Theorem

Let $V$ be a vector space with two norms $||\quad||_{1}$, $||\quad||_{2}$, making $V$ a complete normed vector space. Assume $\exists C$ (constant) such that: $||v||_{2} \leq C||v||_{1}, \forall v ...
1
vote
2answers
40 views

How do I show $(a+b)^K=\sum _{n=0}^K \binom{K}{n} b^n a^{K-n} $ without using recurrence?

We already know that we can represent this binomial as the following: $$(a+b)^K=\sum _{n=0}^K \binom{K}{n} b^n a^{K-n};$$ where $\binom{K}{n} = \frac{K!}{n! (K-n)!}$ My question here is :How do I ...
1
vote
2answers
65 views

Show that $\sqrt{6+4\sqrt{2}}-\sqrt{2}$ is rational using the rational zeros theorem [duplicate]

Let $r=\sqrt{6+4\sqrt{2}}-\sqrt{2}$, then $r+\sqrt{2}=\sqrt{6+4\sqrt{2}}$. Squaring both sides, we get $$r^2+2r\sqrt{2}+2=6+4\sqrt{2}$$ which is the same as $r^2-4=2\sqrt{2}$. Squaring both sides ...
2
votes
0answers
45 views

Is my logic on general Proof-Solving techniques correct?

I've just recently started working through proofs for what's really the first time in my life. Throughout high school, and thus far in college I've never really had to prove things too often, and if I ...
8
votes
5answers
127 views

Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem

What I've done so far: Let $$r = \sqrt{4+2\sqrt{3}}-\sqrt{3}.$$ Thus, $$r^2 = 2\sqrt{3}-2\sqrt{3}\sqrt{4+2\sqrt{3}}+7$$ and ...
0
votes
2answers
93 views

Proving there exist an interval and a number $p$ where $f(x) \leq x^p$ holds

We have a function $f:\mathbb{R}\to [0,1]$, where $f(x)=0$, for $x\leq 0$, and $f$ is a right-continuous function. How can we prove that there exist a number $0 \lt p \leq 1$ and a number $0\lt a$, ...
0
votes
3answers
52 views

Prove $1^3+2^3+…+n^3 = (1+2+…+n)^2$ for all positive integers $n$. [duplicate]

My approach is to solve this by induction. Base case: $n=1$ $1^3 = 1^2 = 1$ Inductive Step: Suppose that $1^3+2^3+...+n^3 = (1+2+...+n)^2$ holds for all positive integers $n$. We use that to ...
28
votes
6answers
2k views

When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
1
vote
1answer
21 views

Operator norm and equivalent definitions

From the definition of the operator norm, we have: $||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$ If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have ...
0
votes
2answers
85 views

Show that $A \lor B ⊢ B \lor A$

Prove the following derivability claim using only our primitive rules: $A \lor B ⊢ B \lor A$ I know this can be attributed to the commutative property, but I'm not exactly sure how to prove this ...
0
votes
1answer
25 views

Proving that $A \Delta C \subset (A \Delta B) \cup (B \Delta C)$

For two sets we define $A \Delta B = (A \cup B) \setminus (A \cap B)$ (symmetric difference). Problem: Proof that \begin{align*} A \Delta C \subset (A \Delta B) \cup (B \Delta C). \end{align*} ...
1
vote
2answers
64 views

Proving that $\log_2 7$ is irrational

Prove that $\log_2 7$ is irrational. Book solution: Suppose $\log_2 7$ is rational. Then $\log_2 7=a/b$, where $a$ and $b$ are integers. We may assume that $a>0$ and $b>0$. We have $2^{a/b}=7$, ...
1
vote
0answers
35 views

Cauchy's integral formula $\Rightarrow$ Green's theorem

I have seen some articles proving Cauchy's integral formula using Green's theorem, but I haven't seen the converse. (And I don't like this approach since, to completely prove each theorem, one should ...
7
votes
0answers
133 views

Conjectured compositeness tests for $N=b^n \pm b \pm 1$

How to prove that these conjectures are true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ...
0
votes
1answer
66 views

How to show that we reach $1$ at an odd or even turn without brute force

Consider the following challenge between two players A and B. They are given the initial terms $a_0= 3^{2014}$ and $b_0= 15^{4028}$ of two sequences, and the scope is to reach $1$ before the other, ...
-1
votes
0answers
45 views

Are all the four roots of this polynomial, eigenvalues of $A$?

Let $A$ be a real symmetric block matrix of order $12$. I guess $x=(\alpha_{1},\alpha_1..\alpha_1,\alpha_2,... \alpha_{2},\alpha_n ..., \alpha_{n})\not=0$, to be an eigenvector and go for solving ...
-4
votes
1answer
22 views

Proving that Changing the Index of the Lower Bound of a Convergent Infinite Series Does Not Affect the Convergence [closed]

Would someone help me in proving that the following theorem is true? Let $j$ be a positive integer. Show that $$\sum\limits_{k=0}^\infty a_{k} \quad\textrm{ converges iff }\quad ...
3
votes
3answers
48 views

Use class algebra to prove the following: If A∩B = ∅ and A∪B = C, then A = C-B

I'm having a bit of trouble proving the following. If A∩B = ∅ and A∪B = C, then A = C-B My initial attempt is to prove it directly, however, I believe I'm assuming the consequent, namely, A = C-B, ...
2
votes
2answers
76 views

Is it possible to find such a $f$?

I search a continuous function $f : [0,+\infty[ \to \mathbb{R}$ such as : $\lim \limits_{x\to +\infty} \frac{1}{x} \int \limits_{0}^{x} f(t)\mathrm{d}t=\pm \infty$ and ($\lim \limits_{x\to ...
3
votes
3answers
78 views

Prove that if $\mathcal{F} \subseteq \mathcal{G}$ then $\cup\mathcal{F} \subseteq \cup\mathcal{G}$

Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. Prove that if $\mathcal{F} \subseteq \mathcal{G}$ then $\cup\mathcal{F} \subseteq \cup\mathcal{G}$ My attempt: Given $\mathcal{F} ...
1
vote
2answers
80 views

proof of number of prime factors of $n$

Given an integer $n$ between 1 and 1000000, how do you directly prove that $n$ has at most 19 prime factors (with multiplicity)? I'm quite stuck on how to do this. I can understand the base case ...
1
vote
1answer
34 views

Extending a Field Monomorphism

In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following. Let $\sigma: F \rightarrow L$ be a field monomorphism ...
3
votes
4answers
74 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
3
votes
5answers
62 views

Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
0
votes
3answers
91 views

Please Help me understand this proof

DOUBT What i didnot understand is from where it is written our new goal means there exists a... I didnot understand how there exists word popped up here and why the new givens are written as they ...
0
votes
1answer
52 views

Could someone give a detailed (yet elementary) proof for Jensen's inequality?

I want to prove that Suppose there is a function $f:[a,b] \to \mathbb R$, and there are $x_i \in [a,b], w_i \gt 0 $ for $i=1,\dots,n$ such that $\sum_{i=1}^nw_i=1$, then if the function is convex, ...
2
votes
5answers
53 views

Showing that $\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$ for all $n\geq 1$

Show that $$\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$$ for all $n\geq 1$ I need this in order to complete my proof that $1 + \frac{n}{2} \leq H_{2^n}$, but ...
0
votes
3answers
27 views

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$ I'm unsure how to proceed with proving this. Basically my idea is as ...
0
votes
1answer
57 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
2
votes
0answers
29 views

Proofs by analysing games

I recently read the following article giving a novel proof of the uncountability of $\mathbb{R}$ by analysing a particular game, amongst other results. ...
6
votes
4answers
57 views

Proving $\sum_{i=1}^n 2^i = 2^{n+1} - 2$ using strong induction [duplicate]

I just started learning proof by induction in class, but got a problem requiring proof by strong induction. Here is the problem. Prove by strong induction: $$\sum_{i=1}^n 2^i = 2^{n+1} - 2$$ ...