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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34 views

What is the difference between these two proofs?

I am studying proofs, and I am stuck thinking about the logic behind these two propositions: 1) Let $x \in\mathbb Z$, if $x$ has the property the for all $m \in\mathbb Z$ $mx = m$, then $x=1$. 2) Let ...
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1answer
31 views

Proofs regarding radius of convergence

Consider $\sum_{n = 0}^{\infty} a_{n}x^{n}$ with radius of convergence R. a) Prove that if all $a_{n} \in \mathbb{Z}$ and if infinitely many of them are nonzero, then $R \leq 1$ Proof: Assume $R ...
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3answers
69 views

Proving a theorem using another theorem

I need help proving this Theorem 1: If (k, b) = 1 and k|ab, then k|a. I was looking at another theorem which may help me prove the above, but it doesn't make much sense to me. Theorem 2: If a and b ...
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0answers
24 views

The cartesian product of compact sets is compact

my question this time is How to prove the following: Let $x \in \mathbb{R}^{n}$ and $E \subset \mathbb{R}^{m}$ compact, prove that $\{x\} \times E$ is compact and form here how to conclude that if ...
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1answer
141 views

Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or ...
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1answer
47 views

Divisibility by zero..

I have two propositions to prove: 1) $0$ is divisible by every integer. Here is my strategy: Proof: Let $j,m\in\mathbb Z$. Now, we multiply to get $0$: $j \cdot m = 0$. Since $0$ can also be written ...
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2answers
27 views

Given that the multiplication of two even numbers gives an even number

I am given the following proposition: If $m$ and $n$ are even integers, then $mn$ is also an even integer. This is my strategy: An integer $m$ is said to be even if it is divisible by 2 (integer). ...
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1answer
18 views

Question about proving the following proposition

I am learning proofs and have the following proposition: For all $m\in\mathbb Z$, $m\times 0 = 0 = 0\times m$ Is it asking me to first prove that $m\times0 = 0$? Because $m\times 0 = 0 \times m$ is ...
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0answers
25 views

Does anybody know of the proof of conditional entropy in general

I was wondering if someone could provide a proof of conditional entropy in general or with two and three variables or a place where I could find it. I am having trouble with some of the algebra and ...
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1answer
63 views

The intersection of non-empty ,nested, and compact sets, subsets of $\mathbb{R}^{n}$ is not empty

Well I need to show that if $A_1\supset A_2\supset\ldots$ are compact and nonempty in $\mathbb{R}^{n}$, then $\bigcap_i A_i \not=\emptyset$, but I think there is something wrong, because $\bigcap_i ...
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1answer
13 views

Show that his cover does not have any finite sub covers

My question is how to prove that the following cover does not have a finite sub cover : $$A_{n}=\{x \in \mathbb{R}^{n}: \frac{1}{2n}<|x|<\frac{3}{2n} \}$$ for the pointed ball $B_1^*(x) = \{ ...
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1answer
31 views

Compactness in $\mathbb{R}^{n}$

My question is once we have that if $x \in \mathbb{R}^{n}$ and $E \subset \mathbb{R}^{m}$ then $\{x\}$ x $E$ is compact how to conclude that if $E \subset \mathbb{R}^{n}$ and $F \subset ...
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0answers
46 views

Is 3-regular graph on 7 vertices disconnected?

Can 3-regular graph on 7 vertices be disconnected? Can you explain why it must be disconnected. 3*7/2 is not integer because of that it can't be drawn. But can i draw disconnect one? Thank you for ...
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2answers
22 views

Proving via Big O addition

I know that it's valid to add and mutltiply functions in Big O, although I haven't seen a proof why. As such I think this is a valid starting point. However, I have no idea how to progress and any ...
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0answers
19 views

Defining the sum in the integers rigorously

My question is: Am I right with this definition of the sum in the integers?. Well in class we covered the definition of the integers knowing the natural numbers as follows: $$\mathbb{Z}=\{(a,b):a ...
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1answer
52 views

Finding a natural transformation between the double dual functor and the trivial one

My question is about to find a natural transformation from $t$ to the trivial functor, and $t$ goes from the following: $$\mathbb{V} \to \mathbb{V^{*}} \to \mathbb{V}$$ then $t$ has to go from ...
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2answers
42 views

Calculus partial derivatives problem; how can I prove this relationship?

Can anyone help me with this proof, I've attached my working out so far: Show that: $\phi =Ae^{-\frac{kt}{2}}sin(pt)cos(qx)$ satisfies the equation $$\frac{\partial \phi^{2}}{\partial ...
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1answer
74 views

Help with Infimum and Supremum in inequality.

I have a problem let s2 = {x in R : x > 0}. Does s2 have lower bound, upper bound? Does inf(s2) and sup(s2) exist? I understand the that the lower bound is 0 while there is no upper bound. I think I ...
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1answer
66 views

Proving that one function is big o of another?

I'm working through a big-O problem and have the intuition to know the answer, but don't feel comfortable in my proof. I need to prove from definitions (i.e. proving that there exists two constants ...
3
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1answer
40 views

Does this proof account for all integers? I am new to proofs.

I am learning proofs and, have the following statement: Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m+x=m$, then $x=0$. Here is my strategy: ...
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2answers
58 views

Find the limit of function $f_{n}(x) = nx^{n}$

Show that $\lim_{n \rightarrow \infty} nx^{n} = 0$ for $x \in [0, 1), n \in \mathbb{N}$ $\lim_{n \rightarrow \infty} nx^{n}$ $\Rightarrow \lim_{n \rightarrow \infty} n \cdot \lim_{n \rightarrow ...
2
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1answer
35 views

Show that this is indeed a functor

my question is about how to prove that the following is indeed a functor, I will describe it as follows: Let $\mathbb{V}$ denote the category of F-vector spaces.Given $V \in Ob(\mathbb{V})$ let: ...
3
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1answer
55 views

Do absolute convergence of $a_n$ implies convergence of $K_n=\frac{1}{\ln(n^2+1)}\sum_{k=1}^{+\infty}a_k\frac{3k^3-2k}{7-k^3}\sin k$?

The problem asks us to decide whether the following statement is true: Let $\{a_n\}_{n\geq1}$ be any absolutely convergent sequence. Does that imply that the sequence: ...
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3answers
127 views

Image of the union and intersection of sets.

Let $f:X\to Y$ be a function, and let $\{S_{i}:i\in I\}$ be a family of subsets of $X$. Then, $$f(\bigcup_{i \in I}S_i) = \bigcup_{i \in I}f(S_i).$$ The case where $f(A\cup B)= f(A)\cup f(B)$ is ...
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4answers
71 views

why these statements are true

I'm reading this book How to Read and Do Proofs. In "Preface to the Student" and "Preface to the Instructor", the author claims to keep the material simple and easy to be followed by students. I was ...
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0answers
33 views

Prove without method of contradiction that there exists a real number less than every positive real number that is positive

This question was asked before for proof by contradiction and which got me into thinking whether i could prove it without using a contradiction Original problem statement is here Prove by ...
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2answers
161 views

Prove that if a and b are positive real numbers, then a + b $\geq$ ab

As the title states, the question is: Prove that if a and b are positive real numbers, then $a + b \geq ab$ For this proof, I'm supposed to prove by contrapositive. So, I get this as a general ...
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1answer
19 views

Associative rule of subspaces and vector spaces

If $U, V, W$ are subspaces of a vector space $X$ and if $U$ is a subspaces of $W$, then $$(U+V) \cap W = U + (V \cap W)$$ Do we need also that $V$ is a subspace of $W$? I can't see how to prove ...
2
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3answers
71 views

I don't understand the difference between the following two statements

I am studying a book on proofs and there are two statements that I don't understand the difference: 1) Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m + x ...
1
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3answers
64 views

Proving that $\{p\to q, p\to \neg q\}\Rightarrow\neg p$

Prove the following: $\{p\to q, p\to \neg q\}\Rightarrow\neg p$, that is, prove that $\neg p$ is a tautological consequence of $\{p\to q, p\to \neg q\}$ (Note: I write $0,1$ instead of $F,T$.) ...
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2answers
76 views

Proving: if $|a|<\epsilon \forall \epsilon>0$ then $a=0$ using a direct proof

I am asked to prove: if $|a|<\epsilon,\forall \epsilon>0$, then $a=0$ I can prove this as follows. Assume $a \not= 0$ I want to show then that $|a| \geq \epsilon$ for some $\epsilon$ We ...
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1answer
41 views

Proving equivalence of two definitions

Hi all I am intersted in proving the equivalence of the following two definitions of pseudomontoncity: Let $V$ be a reflexive Banach space and $K \subset V$ closed and convex. Definition 1: $A: V ...
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1answer
74 views

Puzzle : Truant List of Statements

I was working my way through some puzzles in Discrete Maths by Rosen, when I came across the following question: The $n^{th}$ statement in a list of 100 statements is : "Exactly $n$ of the ...
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1answer
52 views

Please help with the following questions or at least one of them.

Problem 1. Let $f \colon X\to Y$ be a function from the set $X$ to the set $Y$. For a subset $A\subset X$, let $f_*(A)=\{ y\in Y | \exists x\in A\text{ such that }f(x)=y\}$. Prove the following: ...
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1answer
36 views

Proving that the integral of $\cos^m(x)\sin{(nx)}$ between $0$ and $\pi$ is zero

I've been doing a question that initially asks to derive a reduction formula for the indefinite integral of $\cos^m(x)\sin{(nx)}$ then the next part asks to prove that: ...
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2answers
33 views

Proving an Inequality Involving the Modulus of the Difference of Moduli

Prove the following inequality and give necessary and sufficient conditions for equality. $\left| |z|-|w| \right| \leq |z-w|$ for complex numbers $z$ and $w$. I have the following: By definition ...
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1answer
29 views

Proving a number doesn't divide another and proving $lcm$ using the definition

Say I have two integers $a,b$ and I want to prove that $a\not \mid b$ or $ak\neq b$, do I have to take two adjacent $k$s such that $ak_1 < b$ and $ak_2> b$? Is there another way? Another ...
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4answers
53 views

Proving that if $8\mid (n^2+2n)$ then $2\mid n$

Let $n\in \mathbb N$ prove that if $8\mid (n^2+2n)$ then $2\mid n$. From the given, there exists $k\in \mathbb N$ such that $8k= (n^2+2n)$, take $k=1$, and we get $2\cdot 4 = n(n+2)$. Now my ...
3
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2answers
123 views

Help demonstrate how to arrive at the implication of some given inequalities and equations

Given: $0<x<y<1$ $z=x+y$ $x=u$ $y=z-u$ $0<u<z-u<1$ I need to show that this implies: If $0<z<1$, then $0<u<\frac{z}{2}$, and If $1<z<2$, then ...
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4answers
64 views

Proving $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ [duplicate]

Prove $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ I think a proof by contradiction is the easiest in this case, so we have: $\forall x\in \mathbb R :x>0\wedge \neg(x+\frac ...
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2answers
151 views

If there exist integers m and n such that am + bn =1 and c≠± 1, then c does not divide a or c does not divide b

Prove that for all integers a, b, and c. If there exist integers m and n such that am + bn =1 and c cannot be equal to 1 or negative 1, then c does not divide a or c does not divide b. This is the ...
3
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2answers
39 views

Use Induction to prove $\forall m,n \in \Bbb Z_{\ge 0}, 1 +mn \leq (1 + m)^n$

Use Induction to prove: $$\forall m,n \in N, 1 +mn \leq (1 + m)^n$$ for integers $m,n\ge 0$. My biggest problem with this proof is ...
0
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0answers
16 views

Proof verification (limit superior)

Could please somebody verify the proof? $x_n$ is a sequence of real numbers $\lim_{n \to 0} x_n \ne x$, show that $\exists \epsilon >0$, $\lim \sup_{n \to \infty} |x-x_n|>\epsilon$. Proof by ...
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0answers
56 views

rigorous proofs in pure math

I cant figure out how to start proofs not even easy ones. what can I do to fix this. I know I can practice, but nothing seems to help. I don't even know what assumption to make or what to start with
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1answer
115 views

Properties of the inverse of unit (lower) triangular matrix

Is there any special properties about the inverse of a unit lower triangular matrix? I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$ where $L$ is a unit lower triangular matrix ...
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1answer
42 views

Understanding proof of The Ratio Root test

Now this is how I reason. I first try to identify which method that is used to give the proof. I am however so bad at identifying if there are any "hidden" quantifiers in the text. (if there are ...
1
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3answers
123 views

Proof that the graph of a linear function and its inverse cannot be perpendicular.

I am refreshing my high school maths and got an exercise to proof that the graph of a linear function and its inverse cannot be perpendicular. Below is my proof. A linear function is a straight ...
1
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2answers
49 views

Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
0
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1answer
31 views

proof detail concerning bijection between a set and its power set

Theorem: If $X$ is a set, then $X$ is not equivalent to its power set. Proof: suppose for a contradiction that $f:X\to P(X)$ is a bijection. Define $B:=\{x \in X, x\not\in f(x)\}$. Because $f$ is ...
1
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1answer
89 views

Wheel Graphs and Dimension of Embeddings

I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I ...