For questions about the formulation of a proof, not about the mathematics behind it.

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0
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2answers
68 views

A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
0
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2answers
77 views

Prove integral equality $ \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx $ [closed]

How can I prove the following claim for any given continues function: $$ \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx $$ Thanks!
0
votes
2answers
34 views

Hints on showing that a metric space is complete

Let $C[0,K]$ be the space of all continuous real valued functions on $[0,K]$ for $K>0$ and $L\geq0$, equipped with the metric $d$ defined by $$d(f,g)=\sup_{0\leq k\leq K}e^{-Lk}|f(k)-g(k)|.$$ I ...
6
votes
1answer
566 views

Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
6
votes
4answers
519 views

Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
1
vote
1answer
34 views

Did I prove and disprove the following statements correctly?

Let $A = \left\{x \in \mathbb{Z} \mid \exists a\in\mathbb{Z}: x = 6a + 4\right\}$ and $B = \left\{y \in \mathbb{Z} \mid \exists b\in\mathbb{Z}: y = 18b - 2\right\}$ and $C = \left\{z \in\mathbb{Z} \...
0
votes
0answers
33 views

help with solution using mengers theorem

to show: for a $3$ regular graph $G$ we have: edge connectivity $=$ vertex connectivity . attempt: take a minimal seperating vertex set $X$ of $G$ with $|X|=:k$. Then $G \backslash X$ has ...
2
votes
4answers
65 views

My proof that if $P(A) \subseteq P(B)$, then $A \subseteq B$

I'm not sure if my proof is sound. Here it is: Assume that $P(A) \subseteq P(B)$, so any subset C of A is also a subset of B. Therefore, any element in C is also an element of A, and by the same ...
3
votes
1answer
33 views

Not understanding the proof that there is no surjection from a set to its powerset

Here is the question: If a set, $A$, is finite, then $|A| < 2^{|A|} = |P(A)|$, and so there is no surjection from set $A$ to its powerset. Show that this is still true if $A$ is infinite. ...
1
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1answer
48 views

Help in proof: a connected graph is $k$ edge connected iff all blocks are

Attempt: we know that the edge set of $G$ is the union of those of it's blocks (maximal connected subgraphs of $G$ not having a cut vertex), any two of them touching in at most one vertex. If all ...
5
votes
1answer
97 views

New Proof of Pythagorean Theorem (using inscribed circle)?

I was solving an easy problem for fun when I stumbled onto this, and was wondering if this was a correct and possibly a new proof of the Pythagorean Theorem. Given right triangle $\triangle ABC$, and ...
1
vote
1answer
63 views

Battle between Intuition and Rigor in Mathematics in the Context of Computers [closed]

I understand the reason behind the inclusion of rigor in mathematics: to ensure that all new theorems, axioms, and postulates are 100% correct. However, with the advent of computer simulations and so ...
0
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2answers
25 views
19
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14answers
2k views

How are proofs formatted when the answer is a counterexample?

Suppose it is asked: Prove or find a counterexample: the sum of two integers is odd The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which ...
0
votes
1answer
22 views

How can you prove a quadrilateral given a diagonal and segments going from the vertices to the diagonal (picture)?

I'm having quite a bit of trouble with this proof. The angles formed by the segments between diagonals and vertices are 90 degrees, and the vertices on the diagonal and the diagonals' vertices are ...
0
votes
2answers
20 views

Why the set of points satisfying (2x-x²-y²)*(x²+y²-x)>0 is the same of the set with condition ((2x-x²-y²)>o and (x²+y²-x)>0)?

Why the set of points on $\Bbb R^{2}$ satisfying $(2x-x²-y²)(x²+y²-x)>0$ is the same of the set of all elements on $\Bbb R^{2}$ with condition $((2x-x²-y²)>0 $ and $(x²+y²-x)>0)$, i think ...
-1
votes
0answers
11 views

Formal proof on continuity of multivariable function.

Let $B(o,r) = ${$(x,y)\in \Bbb R: \left\lVert (x,y) \right\rVert<r$}$ $ , to some r>0 and the norm is an euclidean norm, let $f(x,y)$ -> $L$ as (x,y) -> (0,0), with f : $B(o,r)$ -> $\Bbb R$. ...
1
vote
2answers
19 views

Is this collection of function uniformly equicontinuous? Hints on the proof.

Let $f_n(x)=\frac{1}{n}\cos(e^{nx})$ for $n\in\mathbb{N}$ be a sequence of functions for $x\in[0,1]$. Is it true that $\{f_n\}$ is a uniformly equicontinuous collection of functions? My attempt so ...
9
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0answers
52 views

What to keep in mind when attempting proof of basic properties of divisibility/what techniques are useful/what's the intuition for showing them?

So I am currently trying to prove some basic divsiibility relations, as follows. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$. If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$. ...
0
votes
1answer
16 views

How to show a set is compact using sequential compactness definition?

Let $l^{\infty}$ be the vector space of all bounded sequences $x=(x_n)$ of real numbers with the norm $||x||=\sup_{n\in\mathbb{N}}|x_n|$ and $l^{\infty}$ is complete. I am trying to show that the set ...
1
vote
1answer
38 views

Existence proof of subspace of projections from V to W

Hi I am having some trouble with this proof. I don't know in which direction to start. I was thinking of using the inner product over C and applying the conditions, then showing that the subspace is ...
1
vote
1answer
48 views

Principle Mathematical Induction Product Notation Question?

$$ \prod_{i=1}^n \left( 2i-1 \right) = \left( \frac{(2n)!}{n!2^{n}} \right) $$ My question for this proof is that I am trying to understand product notation and how to factor this question to its ...
0
votes
5answers
44 views

Proof by Mathematical Induction for all natural numbers n.

$1^3 + 2^3 + \cdot \cdot \cdot+ n^3 = $ $[ \frac{n(n+1)}{2}]^{2} $ $\text{My question for this problem is that I got stuck at a certain point}$ $\text{and I do not know where to go. This is what I ...
1
vote
2answers
46 views

Help determining if a finite subset of $\mathbb R$ is closed and bounded.

If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
2
votes
3answers
67 views

The limit of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ as $ (x,y) \to (0,0)$

In order to prove that the limit as $(x,y)$ approaches to $(0,0)$ of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ is equal to $0$ is wanted to proof: for ever $\beta\gt0$ exists some $\delta\gt0 $such that ...
0
votes
0answers
19 views

About continuity of scalar fields.

Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
-1
votes
0answers
28 views

Determine if the following sentence is a proposition

$2^{101}-1 $ is a prime. Besides being a prime and not being a prime, is there any other case the answer could be? If there isnt a third case, Then is a proposition correct? but if there is then is ...
0
votes
1answer
35 views

Prove that $x=5\cos(6x)$ for some $x$ in the interval $\left(\frac{\pi}{6},\frac{\pi}{3}\right)$

Prove that $x=5\cos(6x)$ for some $x$ in the interval $\left(\frac{\pi}{6},\frac{\pi}{3}\right)$ using the the IVT I'm not entirely sure how to prove this but I set it up this way: $f(x)=x-5\cos(...
0
votes
1answer
17 views

Sandwiched sequence converge to same limit in $\omega_1$

I am stuck on a question that might need a trick to crack, any help is appreciated Problem statement Let $(a_n), (b_n)$ be sequences on $\omega_1$ as a topological space, such that $a_n \leq ...
1
vote
1answer
42 views

Prove using the epsilon definition

I'm trying to prove the below using the $\epsilon$ definition: $\epsilon$-definition: $\;\left|s_n-s\right| \lt \epsilon$ $\lim \limits_{n \to \infty}\frac{(-1)^n\cos \sqrt{n}}{\sqrt[3]{n}}=...
2
votes
3answers
48 views

How to show that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set.

How can it be proved that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set? I tried to prove directly the aforementioned statement. Without success I tried to prove that the image ...
0
votes
0answers
20 views

After the midnight when all trhee clock-hands are in same direction (superposed) again?

in order to solve this question: "After the midnight when all trhee clock-hands are in same direction (superposed) again?" that appears to be simple, probably is, but i could not give a answer ...
1
vote
3answers
41 views

Prove that there exists n consecutive composite numbers

I'm asked to prove that there exists n consecutive composite numbers. This is what I've come up with. $$n! + 1 = (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \dotsc \cdot n) + 1 $$ is a prime number ...
3
votes
2answers
84 views

Prove that $\lim \limits_{x \to 5}\left(4x^2-7\right)=93$

So I first need to determine the limit and then prove it: $\lim \limits_{x \to 5}\left(4x^2-7\right)$ So $L=93$ And thus $\left|f(x)-L\right|=\epsilon$ and $\left|x-c\right|=\delta$ Plugging ...
0
votes
1answer
22 views

How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
3
votes
1answer
28 views

Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$

Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$ This is about Riemann integration. I dont know how to ...
2
votes
1answer
40 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
0
votes
1answer
35 views

Proving that the following is a convex subset

It is given that $I(S)=\{z|\exists x,y \in \mathbb{R}$ such that $(x,y,z)^t \in S$}$\subseteq \mathbb{R}$. How do I show that I(S) is a convex subset? We know that $S \subseteq \mathbb{R}^3$ is ...
0
votes
1answer
39 views

Proof of non-constant analytic functions

Use the following theorem: "A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D" to show that if $f(z)$ ...
0
votes
1answer
35 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
1
vote
0answers
33 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
0
votes
1answer
24 views

Prove if $x_1,…,x_n$ are natural numbers with $n\geq2$ then $x_1x_2…x_n$ is odd iff $x_i$ is odd for all $i$, $1\leq i\leq n$

I am not sure if Im on the right track here but if any one could help out I would greatly appreciate it. Prove if $x_1,...,x_n$ are natural numbers with $n\geq2$ then $x_1x_2...x_n$ is odd iff $...
0
votes
0answers
29 views

How to do proofs involving sets

I have just recently started preparing for a course I will be taking next year, but I have very limited knowledge as it relates to proofs. It seems as though the only proofs I am slightly familiar ...
1
vote
1answer
33 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
1
vote
2answers
51 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
0
votes
0answers
35 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
1
vote
1answer
77 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
0
votes
2answers
41 views

binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
1
vote
1answer
35 views

Is it sufficient to prove that a function is an open map by looking at the basis element?

I am trying to prove that the projection map $\pi_X:(X, T)\times (Y,J) \to X$ is an open map But I don't know if I can use the basis element directly, so my proof is quite round about and lengthy ...
1
vote
1answer
16 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...