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Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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Proving sum of first n non negative integers is n(n+1)/2 using Proof by contradiction method

How could I prove using "Proof by contradiction" method the following: Sum of first n non negative integers is: n(n+1)/2
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1answer
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If $B_{n}=\bigcup _{i=1}^{n}A_{i}$, prove that $\bar{B}_{n}=\bigcup _{i=1}^{n}\bar{A}_{i}$.

If $B_{n}=\bigcup _{i=1}^{n}A_{i}$, prove that $\bar{B}_{n}=\bigcup _{i=1}^{n}\bar{A}_{i}$, where $\bar{E}$ is defined as the closure set of $E$, which is the union of $E$ and its limit points $E'$. ...
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0answers
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Show that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=\lim_{n\rightarrow\infty}\sum_{v=0}^{n}\frac{x^v}{v!}:=\sum_{v=0}^{\infty}\frac{x^v}{v!}$

I have a hard time to understand a certain part of the proof below Let $e_n:=(1+\frac{1}{n})^n$ and $s_n:=\sum_{v=0}^{n}\frac{x^v}{v!}$ According to the binomial Theorem we have $$e_n=1+n\cdot\frac{...
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5answers
681 views

Writing a proof

I am stuck at showing how if: $P \rightarrow Q$ then this implies ($Q \rightarrow R) \rightarrow (P \rightarrow R)$ I know that if we assume $P$ is true then $Q$ also must be true. Therefore if $Q$ ...
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0answers
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Clarification on the method used to prove the existence of $\sqrt{2}$ in $\mathbb{R}$

Let $A=\{x\in \mathbb{R}:x^2\leq{2}\}$ with $\sup A=\alpha$. When proving that the $\sqrt{2}$ exists in $\mathbb{R}$ using the method of contradiction, do we show that a contradiction arises for the ...
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1answer
28 views

Why do these two elements exist and satisfy this equation?

From Convex optimization by Boyd and Vanderberghe: In the below red box, why do there exist $y1,y2$ such that the equation follows? Doesn't it depend on what the domain of $f$ and what the function $...
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0answers
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Proof that $1^\alpha$ is dense on the unit circle

We take an irrational, real $\alpha$. I have got as far to show that $1^{\alpha}=e^{log(1^{\alpha})}=e^{\alpha ln(1)+i\alpha arg(1)}=e^{i\alpha arg(1)}=e^{i\alpha 2\pi k},k\in\mathbb{Z}$. After ...
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2answers
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2 has a square root in $\mathbb{R}$ - proof explanation

Can someone please help me to understand the steps of the proof below. What do the assumptions become when we use the proof by contradiction in the cases where $s^2\gt{2}$ and $s^2\lt{2}$? Can you ...
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1answer
35 views

Thm: If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one.

"If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one." The proof in the book goes as: Because n is a composite number, we can write $ n = rs$, with $ 1 < r \leq s < n$. ...
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2answers
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Let x and y be real numbers. Prove that if x≤y+ϵ for every positive real number ϵ, then x≤y. Why do we set ϵ = 1/2(x−y)?

I understand in the solution that we are finding the contrapositive of $x ≤ y$ which is $x > y$ but why do we set set ϵ= $1\over2$(x−y) ? Where does the $1\over2$ come in and why do we subtract $y$ ...
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1answer
35 views

Applying FTC to Integral Equation (Spivak)

The following is an exercise from Spivak's Calculus: Find all continuous functions $f$ satisfying $$\int_0^xf(t)dt=(f(x))^2+C$$ for $C\neq 0$, assuming that $f$ has at most one zero. I ...
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3answers
45 views

How to compute this combinatoric sum? [duplicate]

I have the sum $$\sum_{k=0}^{n} \binom{n}{k} k$$ I know the result is $n 2^{n-1}$ but I don't know how you get there. How does one even begin to simplify a sum like this that has binomial ...
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2answers
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Does the following set satisfy the conditions for a vector space?

Question: Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations below. $x + y = xy$ $cx=x^c$ I can prove 8 theorems but I am getting a different ...
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0answers
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proof of totally bounded and complete $\implies$ compact ($1/2^n$ vs $1/n$ radius open balls)

This question has a proof of that fact, and I was thinking if it is necessary to impose $1/2^n$ as the shrinking radius of the balls. Why not just $1/n$?
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1answer
24 views

Are the set of even functions a subset of continuous functions $C(-\infty,+\infty)$?

My book says that the above set is closed under scalar multiplication and addition. I can't seem to under why the given set is closed under scalar multiplication. For instance, if I take an even ...
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0answers
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Question about parametric equations

Let a curve be paramaterized, so x(t)=g(t) and y(t)=h(t) how do I prove that eliminating the parameter yields a relationships which satisfies the points which (g(t),h(t)) traces (and possibly more)?
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1answer
36 views

Prove function is greater than or equal to 0

Let $f(x)=x^4-2x^3+3x^2-2x+1$. Prove that $f(x) \ge 0$ . My thought is I need factor the function into sum or difference of perfect squares to show it's always non-negative. Any suggestions?
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1answer
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Prove that $A$ is open if and only if $A=\operatorname{int}{A}$

I am quite confused by this statement, let me first post few definition that the paper I am reading, uses. Definition: Neighborhood of a set $A$ of topological space $\mathcal{X}$ is arbitrary open ...
3
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1answer
48 views

Combinatorics problem on k sets and partionning a set of numbers, understanding of the proof

The question I'm struggling with is given here : Remove any number and the remaining numbers can be partitioned into two subsets of equal sum; prove all numbers are equal. The problem is the ...
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0answers
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How to solve (a-b)^n sum inductive proof [duplicate]

For a class I need to prove the following using an inductive proof: $$(a-b)^n=\sum^n_{k=0}\binom{n}{k}a^k*(-b)^{n-k}$$ We are not allowed to substitute c = (-b), which is why it's so difficult and ...
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1answer
53 views

Proving $ 2 $ angles are equal.

Hi, so I am doing a proof but I need some help proving one part of it. I'm having trouble proving that angle $ D'Bi = $ angle $ D'iB $. point $ i $ is the incenter of triangle ABC and D' is the point ...
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0answers
27 views

Semidirect product theorem to translate from german.

I need some help to translate this theorem and its proof, so I wanted to know if anyone could help me doing this. The book is Endliche Grouppen I, by Bertram Huppert, the theorem is the number 17.3, ...
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1answer
17 views

If $B$ is a basis of $V$ and $U\subseteq V$ is linearly independent then there exists a $C\subseteq B$ such that$ \tilde{B}:= U\dot\cup C$ is a basis

It is an application of Zorn's Lemma, It would help me a lot if somebody could explain one part of the proof that I did not understand Why can we describe every element of $B$ as a linear ...
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0answers
16 views

Showing that the quadratic over linear function is convex?

Let $f(x,y) = \frac{x^2}{y}$ be the quadratic-over-linear function. To show it's convex over $\Bbb R \times \Bbb R_{++}$, we have to show that $\nabla^2f$ is positive semi definite.I see that $$\...
2
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1answer
65 views

How to derive the formula for the expected value for maximum of n normal random variables

This is a follow-up question on this one: Expected value for maximum of n normal random variable @RobertIsrael states the following: Presumably the $X_i$ are independent. If $\Phi$ is the ...
1
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1answer
27 views

How to get the cohomology group in the exact sequence? $B_n^* \leftarrow Z_n^* \leftarrow H^n(C;G) \leftarrow B_{n-1}^* \leftarrow Z_{n-1}^*$

In Hatcher page 192, the sequence is extracted $B_n^* \leftarrow Z_n^* \leftarrow H^n(C;G) \leftarrow B_{n-1}^* \leftarrow Z_{n-1}^*$ from the diagram $Z_n^* = \ker \delta_n$, $B_n^* = Img \...
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0answers
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About “Principles of Mathematical Analysis” by Walter Rudin Theorem 3.17(a).

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Definition 3.16: Let $\{ s_n \}$ be a sequence ...
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0answers
55 views

What does “$E$ is not bounded above” mean? I am confused. “Principles of Mathematical Analysis” by Walter Rudin Theorem 3.17.

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Definition 3.16: Let $\{ s_n \}$ be a sequence ...
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4answers
53 views

Prove that for all integers $n$ if $3 \mid n^2$, then $3 \mid n$

Prove that for all entegers $n$ if $3$ | $n^2$, then $3$ | $n$. I figured using contropositve was the best method by using the definition "an integer $k$ is not divisible by 3 if and only if there ...
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1answer
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Subset of a metric space is a metric space.

I have a question? Why is it that every subset of a metric space is a metric space? I mean what if the subset is the empty set, then it can't be a metric space, right? because a metric space is by ...
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1answer
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How do the authors prove that “The relative complement of the Cantor set in $[0,1]$ is dense in $[0,1]$”?

In my textbook Introduction to Set Theory by Hrbacek and Jech, the authors first construct Cantor set: Next they prove The relative complement of the Cantor set in $[0,1]$ is dense in ...
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3answers
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Where is the mistake in this proof that “If the square of a number is even, then the number is even”?

The proof given in This question is incorrect (the proof will be posted at the end for convenience). However, the question seem to address the fact that the statement to be proven is not stated ...
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1answer
33 views

Declaring a metric space as a universe.

For instance, if I wanted to prove that every metric has a particular property, such as every metric space is open in itself. I would first consider and arbitrary metric space (X,d). Therefore ...
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5answers
42 views

Proving whether the set of third degree polynomial is not a vector space?

In my book, it says the above set fails the first axiom. It says if I take two sets $p_1(x)=x^3-x^2$ and $q(x)= 1-x^2$. They are not closed under addition. I can understand why that's true by using a ...
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3answers
77 views

How to you prove $|a|=|-a|$ is true

Let $a \in R$ prove that: $|a|=|-a|$ I am new to proofs so this is my attempt: Case 1: $|a|=|-a|$ $$(a)=-(-a)$$ $$a=a$$ Case 2: $|-a|=|-a|$ $$-(-a)=-(-a)$$ $$a=a$$ Case 3: $|a|=|a|$ $$a=a$$...
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0answers
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On cardinality: $|[0,1]|\ge|\{0,1\}^{\mathbb{N}}|$ and $|\{0,1\}^{\mathbb{N}}|\ge |(0,1)|$

I have already proved the following results: Proposition 1. Let $p\in\mathbb{N}$, $p\ge 2$. Let $x\in [0,1).$ Then exists a sequenece $\{c_k\}\subseteq \mathbb{N}$ such that $0\le c_k\le p-1$...
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0answers
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+50

Why can the test error be written in terms of the training error in this way?

In the below picture encircled in red: If $L_D(h) = E_{z \text{~} D}[l(h, z)]$, Then how is $L_D(h) = E_{S' \text{~} D^m}[L_{S'}(h)] $? I see that $$\large L_D(h) = E_{z \in Z}[l(h, z)] = \...
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3answers
63 views

Proving $\sqrt{2013^{2016}+2014^{2016}}$ is irrational

Prove $\sqrt{2013^{2016}+2014^{2016}}$ is irrational. I've looked at the proofs for proving $\sqrt{\mathstrut 2}$, $\sqrt{\mathstrut 3}$, and$\sqrt{\mathstrut 15}$ irrational. Using proof by ...
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0answers
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Clarification required for the proof of Riemann Lebesgue Lemma

Riemann Lebesgue Lemma Proof. In the proof given above, I came across an unfamiliar notation $g= \sum m_i \chi_{[x_i-1,x_i]}$. What does the $\chi$ mean here? Is it the characteristic function of the ...
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3answers
37 views

Rudin's functional analysis appendix A4 (a)

Quick question about the following theorem: If $K$ is a closed subset of a complete metric space $X$ then the following three properties are equivalent: (a) $K$ is compact (b) Every ...
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2answers
43 views

Show that $\leq$ on $\mathbb{R}$ satisfies the following three properties.

I am new to proofs. Let $A$ be a relation on $\mathbb{R}$. Reflexivity: For all $x\in\mathbb{R}, x \leq x$. Proof: Let $x$ be an element of A. Clearly, $x$ will always equal itself. This clearly ...
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2answers
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How to read $f(K)=\bigcup_{k\in K}\langle f,k\rangle$ and what it means, mathematically

Let $E$ be a real Banach space and let $K$ be a subset of $E$. Suppose for each $f\in E^{*}$, the set $$f(K)=\bigcup_{k\in K}\langle f,k\rangle$$ is bounded in $\Bbb{R}$. Then, $K$ is a bounded subset ...
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1answer
18 views

Convergence of CG method, number of eigenvalues

I am trying to fix this proof of a lemma that I did't correctly write: If $A\in \mathbb{R}^{n\times n}$ symmetric positive definite with $m$ different Eigenvalues $\lambda_j$, then the Conjugate ...
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0answers
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Is the set of all $2 \times 2$ matrices with real eigen values dense in $\Bbb M_2 (\Bbb R)$? [duplicate]

Let $S : = \{A \in \Bbb M_2 (\Bbb R) : \text {both the eigen values of $A$ are real} \}$. Is $S$ dense in $\Bbb M_2 (\Bbb R)$? Please help me in this regard. Thank you very much.
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1answer
27 views

Recurrence relation - second order

I read this book, and I do not understand one thing in proof. The theorem is: Let $s_1, s_2$ be number so that quadratic equation $x^2-s_1x-s_2 = 0$ has exactly one root, $r \neq 0.$ Then ...
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2answers
49 views

Proof in linear algebra/calculus

So I am currently studying Calculus and Linear Algebra and I came across the same concepts that is being applied in a lot of the proofs that I read for Calculus and Linear algebra but not capable of ...
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0answers
24 views

Rudin's functional analysis Theorem 3.18, second part.

Just a follow up to the following two questions: Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded. Theorem 3.18, Rudin's ...
2
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1answer
32 views

Confusion on Additive Property of Supremum and Infimum - Theorem I.33 Apostol Calculus I

There is a great question and answer to the first part of Theorem I.33 of Apostol Calculus "Additive Property" here. I'm hoping someone can verify my attempt at part (b). Apostol provides the ...
0
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1answer
25 views

Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.

Reading through the proof of the following In a locally convex space $X$, every weakly bounded set is originally bounded, and viceversa The trivial part of the proof. Since every weak ...
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1answer
23 views

Proof theorem 3.17 Rudin's functional analysis

Consider the two theorems (3.15 and 3.16) of Rudin's functional analysis: Theorem 3.15: If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$K = \left\{\Lambda \in X^* : |\...