For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1answer
16 views

Prove that every finite A is Dedekind finite.

I'm trying to prove that every finite set $A$ is Dedekind finite. I have to use the theorems: that a set $A$ is finite iff there is a natural number $n$ so that there is a bijection $f: n \rightarrow ...
0
votes
1answer
24 views

Prove that every positive natural number is Dedekind finite.

Prove that every positive natural number is Dedekind finite. I'm trying to prove this theorem with induction. I'm stuck on how I should use induction to prove this theorem.
0
votes
2answers
40 views

Prove that $n<(3/2)^n$ for any $n$ with induction [on hold]

need help with induction with inequality, I suck at it. $n<\left(\frac{3}{2}\right)^n$ for any $n$
2
votes
3answers
179 views

Wheel of Fortune Problem

The Summation formula is $$\sum_{i=1}^ni =\frac{n(n+1)}2$$ How is it that we know the integers $1,2,...36$ appear exactly $3$ times. And why do we multiply the sum by $3$ in the last part of the ...
2
votes
2answers
69 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
2answers
28 views

Proof on limit superior and limit inferior of a set

I understand the result intuitively but how can I prove this? For a given integral $n \ge 1$, let $A_n = \left\{\frac mn \mid m \in \mathbb Z\right\}$. Show that $\varlimsup_{n\to\infty} A_n = ...
0
votes
2answers
46 views

Proving Alternating Series Convergence

Suppose $x_n > 0$ and $\sum_{n=0}^\infty x_n$ is convergent. Prove that $\sum_{n=0}^\infty (-1)^nx_n$ is convergent. Any hints or starting points? So far I figured that I should show that the ...
1
vote
1answer
55 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty]$ be a nonnegative, ...
1
vote
2answers
50 views

Wondering if proof is proper

so I have been working on learning some new math in order to prepare for next year. I have been trying to learn proofs, and doing practice questions however the only problem is there are not answers. ...
5
votes
2answers
115 views

What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
4
votes
1answer
36 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
0
votes
3answers
27 views

Help with understanding definition of divisibility in this case.

I have a proof that shows that if $5 \mid xy$ then $5 \mid x$ or $5 \mid y$. It's pretty clear to me that I can just say that suppose $5 \mid x$, then $x=5a$, where $a$ is an integer. then $xy = ...
2
votes
0answers
14 views

Set with relative complement forms partition

Prove that if $S$ is a set and $ \emptyset \subsetneq A \subsetneq S $ then $\Pi = \{A , S-A \}$ is a partition of $S$. Proposed Solution: Since $ A \subsetneq S$ , we have $S - A \neq ...
0
votes
3answers
27 views

Help to prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$

Prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$ My attempt: $\begin{align} (x,y) \in (A \times B) \cup (C \times D) & \Rightarrow & (x,y) \in (A \times B) \vee ...
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votes
3answers
76 views

Examples of revisited proofs after new theorems are discovered… [closed]

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later? For instance, we know now that Pn+16< Pn+1 occurs infinitely often (where ...
0
votes
1answer
26 views

Need help proving the statement

Assume that D ⊂ N and D ̸= ∅. Prove or disprove using a detailed structured proof, justifying every step: [∀x ∈ D, ∃y ∈ N, y < x] ⇔ [0 ̸∈ D] I have no idea how to prove a statement like that, I'm ...
2
votes
1answer
26 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
0
votes
1answer
44 views

Why are the different ways to write a universal statements equivalent?

Consider the following universal statements: $\forall a \in \mathbb{R}-\{0\}, a^2 > 0$ $\{a \in \mathbb{R} - \{0\}| a^2 > 0 \} = \mathbb{R}-\{0\}$ $a\in \mathbb{R}-\{0\} \Rightarrow a^2>0$ ...
2
votes
1answer
41 views

Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
1
vote
1answer
32 views

Decomposition of a function in positive and negative parts and its integrability

Is it true to say that $\int_\mathbb{R}|f(x)|dx<\infty\Rightarrow\int_\mathbb{R}f(x)=0$?
0
votes
2answers
27 views

Proof for combination using a specific definition

Suppose $n,k,\in\mathbb{Z}$ and $0\leq k \leq n $ prove using the following definition: if n and k are integers then $\binom{n}{k}$ denotes the number of subsets that can be made by choosing k ...
0
votes
1answer
21 views

Decomposition of a function into positive and negative parts and its integrability

1)Is it true that any function can be decomposed as a difference of its positive and its negative part as $f=f^{+}-f^{-}$ or that function should belong to $\mathcal{L}^{1}(\mathbb{R})$. Also if that ...
0
votes
0answers
40 views

wavelets, fourier-transform [duplicate]

We are given that $f\in C^{p} \ if \int_\mathbb{R}|\hat{f}(\omega)|(1+|\omega|^p)d\omega <+\infty$. Now if $\hat{f}$ has a compact support then how $f\in C^{\infty}?$
1
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1answer
53 views

Use of prime symbol in proof writing

Questions: So I looked through my course notes and saw this proof. I understand the content, but I'm confused about the use of the prime symbol. If we say that there is some $j'$ such that it is a ...
7
votes
3answers
89 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
3
votes
0answers
40 views

Prove that $ 1+ \cos A + \cos B + \cos C = 0$. [duplicate]

If $A+B+C=180^\circ$ and $\tan \left[\dfrac{A+B-C} 4 \right] \tan \left[ \dfrac{-A+B+C} 4\right] \tan\left[\dfrac{A-B+C} 4 \right] =1$ then prove that $ 1+ \cos A + \cos B + \cos C = 0$ I ...
1
vote
1answer
26 views

Help to prove the condition that a right half-open interval is not empty

The right half-open interval is defined as: $[a,b) = \{x \in \mathbb{R}|a \le x \lt b\}$ I need to prove: $[a,b) \ne \emptyset \iff a<b$ My attempt: For $\Rightarrow$: $$\begin{align} ...
0
votes
1answer
24 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
1
vote
1answer
32 views

Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
0
votes
3answers
84 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
2
votes
3answers
154 views

Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
2
votes
2answers
58 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
29
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
0
votes
2answers
38 views

need assistance identifying formula and help proving it

This is the identity: $$ \int_{x=0}^\frac{1}{\sqrt 2} \frac{x^{k-1}}{1-x^8}dx = \int_{x=0}^\frac{1}{\sqrt 2} { \sum_{i=0}^\infty x^{k-1+8i}}dx = \frac{1}{{\sqrt 2} ^ k} \sum_{i=0}^\infty ...
1
vote
1answer
24 views

Proving Monotonic Sequence Theorem

A sequence $b_n$ is decreasing and bounded. Prove it it convergent. Proof: Since $b_n$ is bounded, $b_n > L$ where L is the greatest lower bound as per the completeness Axiom. Consider some ...
0
votes
3answers
95 views

Find the problem with this proof.

The following attempts to prove that if $n^2$ is even, then $n$ is even. Suppose $n^2$ is even. Then $n^2$ = 2$k$ for some integer $k$. Let $n$ = 2$m$ for some integer $m$. This shows that $n$ is ...
1
vote
1answer
42 views

Show every dilation is a non-constant linear function.

A dilation of reals is a function $f:\Re \mapsto \Re$ such that for some constant $c\neq0$ one has $|f(x)-f(y)|=c\ast|x-y|$ for all $x,y\in\Re$. Show that every non-constant linear function is a ...
1
vote
0answers
25 views

Proof the Associative Property of H-group

I would like to prove the following proposition: Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a ...
1
vote
1answer
42 views

Provide the Proof for $\forall x\,\bigl ( P(x) \land Q(x)\bigr ) \leftrightarrow \forall x \,P(x) \land \forall x\, Q(x)$

Provide the Proof for $$\forall x\,\bigl ( P(x) \land Q(x)\bigr ) \leftrightarrow \forall x \,P(x) \land \forall x\, Q(x)$$ This is all i got so far: Assume $\forall x \,\bigl( P(x) \land Q(x) ...
1
vote
1answer
58 views

Proof by contradiction: logarithm

I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
0
votes
3answers
40 views

Proof by contradiction proving both numbers are not odd.

I have to do a proof by contradiction: Suppose $a,b,\in\mathbb{Z}$. If $4| (a^2 + b^2)$ then a and b are not both odd. So far I know that I need to prove that if $4|(a^2+b^2)$ then a and b are both ...
0
votes
1answer
37 views

When writing a proof, why do we want to assume a different but equivalent condition given in the proposition? [duplicate]

In the proof for the inductive step, we start by assuming $k \ge 10$. But along the way, the author mentions $k \ge 1$ and $k \ge 7$ to justify the inequality. Why do we bother to do this instead ...
0
votes
3answers
54 views

In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
2
votes
2answers
94 views

Definition of a Limit

Prove that $ \ln\left(e +\frac{1}{n}\right) \to 1$ as $n$ approaches $\infty$. I know I must show $\exists$ $n > N$ such that $\left|\ln\left(e +\frac{1}{n}\right)-1\right|< \varepsilon $ But ...
1
vote
2answers
28 views

Proof that the subset relation is reflexive and transitive

I'm teaching myself set theory, and I'm not sure how detailed I should be when asked to prove things. Here is my proof that $A\subseteq A$ (the subset relation is reflexive): $A \subseteq B$ iff ...
0
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1answer
29 views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
3
votes
0answers
31 views

An undirected graph $G$ can be decomposed into simple edge-disjoint cycles if and only if all of its vertices have even degree.

Research effort: $\rightarrow)$ I think this is relatively easy. $\leftarrow)$ Let $G = (V,E)$, let $w$ be any vertex of $G$, given that all the vertex have even degree, I'm assured that I can ...
1
vote
1answer
35 views

Prove that if sets A and B satisfy this relation, then they have a common element.

I have done the proof by drawing the picture and explaining it by using an example, but how could I start a more formal proof for this example without the use of a numeric example?
1
vote
2answers
29 views

Proving that mean KDR in a videogame is one

This is not related to schoolwork. A friend of mine challenged me to prove that the mean KDR (assuming players can only die at the hands of other players) must always be equal to one. I have gotten ...
1
vote
2answers
43 views

Is there a direct proof of the following?

I have been warned by my Lecture as well as several other sources that while proof by contradiction is useful and is certainly needed in some cases, it is often overused. In a effort to learn, I ...