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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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, ...
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2answers
63 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. ...
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2answers
184 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 ...
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1answer
120 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 ...
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3answers
33 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 = ...
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0answers
32 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 ...
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3answers
83 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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106 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 ...
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1answer
29 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 ...
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1answer
31 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 ...
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1answer
91 views

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

Consider the following universal statements: $\forall a \in \mathbb{R}-\{0\}, a^2 > 0$ $\{a \in \mathbb{R} - \{0\}| a^2 > 0 \} = \mathbb{R}-\{0\}$ $a\in \mathbb{R}-\{0\} \Rightarrow a^2>0$ ...
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1answer
49 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 ...
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1answer
51 views

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

Is it true to say that $\int_\mathbb{R}|f(x)|dx<\infty\Rightarrow\int_\mathbb{R}f(x)=0$?
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2answers
31 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 ...
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1answer
190 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 ...
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1answer
65 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 ...
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3answers
115 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 ...
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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 ...
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1answer
52 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} ...
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1answer
37 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 ...
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1answer
79 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 ...
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3answers
308 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 ...
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3answers
183 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 ...
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2answers
79 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 $\dim W \ge 2$. ...
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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 ...
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1answer
119 views

How to prove the inequality?

Given $0<x<1$, $0<a<b<1$, and $a+b<1$, how to prove $a^x(1-ax)<b^x(1-bx)$? I've tried using $f(x)=x^t(1-xt)$ to do some manipulations (including derivations), but failed.
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2answers
44 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 ...
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1answer
30 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 ...
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3answers
118 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 ...
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1answer
86 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 ...
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0answers
64 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 ...
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1answer
64 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) ...
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1answer
68 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 ...
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3answers
54 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
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1answer
56 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 ...
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3answers
72 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. ...
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2answers
102 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 ...
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2answers
875 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 ...
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1answer
458 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 ...
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167 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 ...
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1answer
41 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?
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39 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 ...
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2answers
50 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 ...
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1answer
151 views

Euler proof of the formula involving factorial?

Let me be formal and write the formula Euler's Formula: Let $a$ and $n$ be nonnegative integers with $a\geq n.$ Then $$n!=a^n-\binom{n}{1}(a-1)^n+\binom{n}{2}(a-2)^n-\binom{n}{3}(a-3)^n+ > ...
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3answers
153 views

Proposed proof of set theoretic result

I am tasked with proving the following: $$ (A - B)\cap (B-A) = \varnothing $$ My Attempt: Suppose there exist a $x \in (A - B)\cap (B-A) $ then: \begin{align*} x \in (A - B)\cap (B-A) &\iff ...
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3answers
130 views

Prove $A = (A \setminus B) \cup (A \cap B)$

Prove $A = (A \setminus B) \cup (A \cap B)$ Logically, this is clearly true. I can explain why: start with $A$, remove all elements in $B$ and then add in any elements in both $A$ and $B$, which ...
3
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1answer
182 views

Prove that $2+2=4$.

Before you might chastise this quesion, I understand that we all know $2+2=4$. But a while ago I just stumbled across this paper which formally proves that $2+2=4$: ...
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98 views

Proof that doesn't exists a rational $s$ such that $s^2 = 6$

Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better ...
4
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1answer
171 views

Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
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2answers
80 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...