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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4
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0answers
65 views
+50

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
1
vote
4answers
22 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
69 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 ...
1
vote
2answers
63 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 ...
4
votes
2answers
45 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...
3
votes
3answers
60 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} ...
3
votes
0answers
61 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
1
vote
1answer
30 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 ...
2
votes
5answers
52 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 ...
3
votes
5answers
55 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
1answer
51 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 : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
3
votes
4answers
69 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 ...
0
votes
3answers
84 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 ...
-1
votes
0answers
26 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, ...
0
votes
3answers
23 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 ...
2
votes
0answers
25 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
56 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: $$\sum_{i=1}^n 2^i = 2^{n+1} - 2$$ ...
1
vote
1answer
87 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of under of the parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
1
vote
3answers
78 views

How to show that a statement in sets is false?

How to show that a statement in sets is false and prove its negation is true? For example I have the exercise: Let's say that $E$ is a non-empty set and $A,B,C$ $\subseteq$$E$.For each $Α,Β,C$ how ...
5
votes
2answers
63 views

Proving that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$

For each $A,B,C$ how would I prove that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$ ? My thoughts are if $x\in (A\setminus C)\cap(B\setminus C)\cap(A\setminus B)$, then $x\in ...
3
votes
0answers
61 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
3
votes
0answers
57 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
3
votes
1answer
299 views

Prove that there is only one way to make a square using all six tangram pieces

I am pretty sure there is only one way to make a square from the six tangram pieces: How can I prove this is the only way respecting all symmetries?
1
vote
5answers
64 views

Concise induction step for proving $\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$

I recently got a book on number theory and am working through some of the basic proofs. I was able to prove that $$\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$$ with the help of the identity ...
0
votes
1answer
531 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
1
vote
1answer
53 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
0
votes
1answer
27 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}$ is a rational square where $ \sigma(k) $ and $k$ both are square?

Is There some one who can show me if there are infinitely many $k$ for which $$\frac{\sigma(k)}{k}$$ is a rational square where $\sigma(k)$ and $k$ both are square ? Note :$\sigma(k)$ is sum ...
0
votes
1answer
26 views

Proof that a real number must occur in the domain between two other real numbers

I saw the question stating that if your speed is $v_0 = 0$ km/h at $t=0$ and your speed is $v_{30} = 20$ km/h at $t=30$, then did you ever had a speed of $v=\pi$? Obviously this is the case as speeds ...
3
votes
2answers
340 views

If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
1
vote
2answers
48 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
1
vote
1answer
30 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
-1
votes
1answer
106 views

Injections, Surjections, Bijections [on hold]

So i was given a question that asks me to determine whether the function is injective, bijective, or surjective. If you answer bijective than determine the functions inverse, domain, and target space. ...
0
votes
0answers
39 views

Help fix this proof.

What is wrong with this proof? I followed the example of the answer to another one of my questions, here Define a general recurrence relation as $$f(x)^2=A(x)+B(x)f(x+n).$$ Substitute the root ...
1
vote
5answers
58 views

Proving that that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$

How might I prove that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$? I am not sure the best place to start other than assuming $x\in(R \setminus S)\setminus T$ and trying to ...
6
votes
1answer
341 views

Application of Jensen's inequality to $x^x+y^y+z^z$

Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$ My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ ...
1
vote
3answers
27 views

Finding all the divisors of $a$ by decomposing it into the product $p^{\alpha_1}_{1} \cdot p^{\alpha_2}_{2} \cdots p^{\alpha_r}_{r}$

I'm trying to prove the following statement regarding the fundamental facts of prime numbers, but I don't really understand the relationship between $a$ and $b$. In order to find all the divisors ...
3
votes
1answer
49 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
-1
votes
2answers
55 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
3
votes
4answers
101 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
0
votes
1answer
76 views

Find the number of flags of different types using induction

A flagpole is $n$ feet tall. On this pole we display flags of the following types: red flags that are $1$ foot tall, blue flags that are $2$ feet tall, and green flags that are $2$ feet ...
0
votes
1answer
43 views

Estimate for the integral using convexity bound

I'm reading the proof of Hardy and Littlewood's theorem in the book Analytic Number Theory, written by Henryk Iwaniec and Emmanuel Kowalski (p. 547): Theorem (Hardy and Littlewood): Let $N_0(T)$ ...
3
votes
4answers
117 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
0
votes
1answer
49 views

A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water $n$ times.

There is a formula in my book for questions of type, A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water. After $n$ such operations what will be the ...
0
votes
0answers
33 views

Demonstrative geometry around the world and its significance.

This is not exactly a mathematical question. I am from Pakistan; and over here students are taught a subject 'demonstrative geometry' (as a part of mathematics) from secondary level education. ...
0
votes
2answers
42 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
2
votes
4answers
58 views

Prove by induction that $\sum_{k=1}^{n} k^3 = \bigg( \sum_{k=1}^{n}k\bigg)^2$ [duplicate]

Show the following for all positive integers using proof by induction: $$\sum_{k=1}^{n} k^3 = \bigg( \sum_{k=1}^{n}k\bigg)^2$$ Base case (n = 1) passes: $1^3 = 1^2$ We assume the following: ...
2
votes
1answer
39 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
-1
votes
0answers
70 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective [closed]

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
0
votes
1answer
25 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
-4
votes
0answers
46 views

prove that for any 2n≥2 and any \a ​1 ​​ ,…,a ​n ​​ ∈N, we have the following: [closed]

So the question I was given goes like this we will introduce a mystery function,P:N→N. We don't know a formula for P (and we won't be able to determine one!) but we do know that P satisfies the ...