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

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1answer
24 views

Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
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1answer
59 views

Prove that $\int_{a}^{b} f > 0$

I am asked to prove that $\int_{a}^{b} f > 0$. we are given that $f$ is continuous on $[a,b]$ $(\forall x \in [a,b]) \; f(x) \geq 0\; $ and $(\exists x_{0}) \in [a,b] \;s.t.\; f(x_{0}) >0$ my ...
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0answers
28 views

Proof Related to the Span in linear algebra

I'm working through a proof in my linear algebra textbook, and I think I am a little stuck. I am trying to prove that if $S$ is a non-empty set of vectors in a vector space $V$, the the set $W_s$ of ...
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0answers
44 views

Is there such a thing as “finite” induction?

I am not sure of the terminology that I am looking for, but I would like to use an inductive proof on the following type of structure. I have something of the form, for every $n \geq 2$ and for any $1 ...
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2answers
19 views

Is this proof correctly written? Show that the sum of two uniformly continuous functions on $A$ is uniformly continuous on $A$

If $f$ and $g$ are uniformly continuous functions in $A$ show that $f+g$ is uniformly continuous in $A$. Proof: because $f$ and $g$ are uniformly continuous on $A$ we can write $$\forall\varepsilon&...
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3answers
102 views

How can I show that $\mathcal{B} = \{(a,b)\subset \mathbb{R}\mid a,b \in \mathbb{Q}\}$ is a countable set?

I know that $\mathcal{B} = \{(a,b)\subset \mathbb{R}\mid a,b \in \mathbb{Q}\}$ is a basis on $\mathbb{R}$. I need to show that $\mathcal{B}$ is countable. How can this be done? Attempt: Take ...
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1answer
23 views

Assume f and g are defined on all of $\mathbb{R}$ and that $\lim_{x\to p} f(x) = q $ and $\lim_{x\to q} g(x) = r $.

(a) Give an example to show that it may not be true that $\lim_{x\to p} g(f(x)) = r$ If we are to assume that f and g are defined on all of $\mathbb{R}$, wouldn't that mean that f and g are ...
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4answers
111 views

Prove $\ \sin(x) < x \ \ \ \forall x \in(0, 2\pi)$

Problem : Prove $\sin(x) < x \ \ \ \forall x \in(0, 2\pi)$ Now I have a possible solution for this, using limits and the first derivatives of $\sin(x)$ and $x$, but I don't feel it's a very ...
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1answer
49 views

If a set K $\subseteq \mathbb{R} $ is closed and bounded, it is compact.

If a sequence is closed and bounded, that means it has a sequence that converges. According to the Bolzano-Weierstrauss Theorem (which I am taking for granted), a converging sequence has subsequences ...
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1answer
23 views

Show that if $K$ is compact and nonempty, then $\sup K$ and $\inf K$ both exist and are elements of $K$.

Show that if $K$ is compact and nonempty, then $\sup K$ and $\inf K$ both exist and are elements of $K$. If $K$ is compact, then by definition it is closed and bounded, and every sequence in $K$ has ...
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2answers
335 views

Dividing a Checkerboard into L-Shaped Regions

In preparation for the GRE Math-Subject test, and honestly for the fun of it, I've been working through a select number of my texts. The first of which is Saracino's Abstract Algebra text. I was ...
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1answer
36 views

Prove that F $ \in \mathbb{R} $ is closed if and only if every Cauchy sequences contained in F has a limit that is also an element of F.

I'm a novice at proofs so I like to write out everything, so please bear with me!. I understand that this is a biconditional statement, and I will have to prove it in the forward and reverse direction....
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2answers
75 views

Why we not check conditions while solving questions?

Note:Down ward problem is just an example to express my question(I know the both solution of problem are insufficient but the first solution is in my 10+2 book and second one is mine which is ...
2
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1answer
39 views

If $2^n-1$ is prime, then n is prime - proof involving the Mersenne primes by counterexample

Let $2^n-1$ be prime. Suppose that $n=p_1p_2\cdots p_s$ is composite. Then we have $2^{p_1p_2\cdots p_s}-1$; call it $k$. If $k$ is prime, then its only divisors are $k$ and $1$. But consider the case ...
0
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1answer
34 views

Show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ using definition

It is well known that $\overline {A \cap B} \neq \overline A \cap \overline B$ I wish to show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ by using the definition (...
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6answers
54 views

Using induction to prove for $n ≥ 1, $ $1 \times 5+2\times6+3\times7 +\cdots +n(n + 4) = \frac 16n(n+1)(2n+13).$

This is a very interesting problem that I came across in an old textbook of mine. So I know its got something to do with mathematical induction, which yields the shortest, simplest proofs, but other ...
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1answer
58 views

How to integrate $\frac{dx}{(x^2+k^2)^m}$, with $m$ positive integer.

How can I integrate: $$\int \frac{1}{(t^2+k^2)^m}\, dt$$ without trygonometric substituition? where $t= (x+(p/2))$ and $k= (1-(p^2/4))$ coming from an equation with complex roots: $x^2 + px + q.$ ...
3
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1answer
49 views

For every integer $m\geq 0$ let $I_m=\int_0^1x^m\left(x^2 -1 \right)^5dx$. Prove that for $m\geq 2$ $I_m= \frac{m-1}{m+11}\,I_{m-2}.$

This is a very interesting problem that I came across in an old textbook of mine. So I know its got something to do with integrals, but other than that, the textbook gave no hints really and I'm ...
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3answers
47 views

How to show the usual topology is finer than co-finite topology on $\mathbb{R}$

I have solved a bunch of problems where the basis is used to quickly deduce which topology is finer than which. However, I do not know the basis of co-finite topology. What is the straight ...
0
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1answer
26 views

A set of discontinuities?

$ \mathbb{Q} $ is countable, so we can list its elements. Let $ \mathbb{Q} = \{r_1 , r_2, ...\}$ Define $f: \mathbb{R} \to \mathbb{R} $ by the following rule: $ f(x) = \begin{cases} 0 & x \...
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1answer
26 views

$\mathcal{T_B}$ is the intersection of all topologies containing $\mathcal{B}$

Let $\mathcal{B}$ be a basis on a set X, and let $\mathcal{T_B}$ be the topology it generates. Show that $\mathcal{T_B} =\bigcap \{ \mathcal{T} \subseteq P(X) \mathcal{T}$ is a topology on X and $\...
2
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1answer
62 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{...
0
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1answer
18 views

Proving the element of a symmetry group $\sigma^i \in S_n$ is of order $n$ and length $n$ only when $(n,i) = 1$

Start with element of $S_n$ as $\sigma^i$ which permutes an element of the set $\{1,2,3,...,n\}$, call it, $a_k \to a_{k+i}$ So $({\sigma^i})^2$ would permute $a_k \to a_{k+2i}$ If $k+i > n$, the ...
0
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1answer
39 views

The bases for the set of all functions f:[0,1]→[0,1]

Let $X = [0, 1]^{[0,1]}$, the set of all functions $f : [0, 1] \rightarrow [0, 1]$. Given a subset $A \subseteq [0, 1]$, let $U_A = \{ f \in X : f(x) = 0 \forall x \in A \}$ . Show that $B := \{U_A : ...
1
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1answer
23 views

proof of $\sin(420º+\alpha) + \cos(60º+\alpha) = \sin(90º-\alpha)$?

I was trying to proof this using the right side, and I'm aware that $\cos (60 + \alpha) + \cos(60 + \alpha)$ it's what I'm really looking for but I can't find a way to proof it. \begin{align} \sin (...
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0answers
23 views

Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
1
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1answer
29 views

The bases for the discrete topology

The collection $\mathcal{B} = \{ \{x\} : x \in X \}$ is a basis for the discrete topology on a set X. If X is a finite set with n elements, then clearly $\mathcal{B}$ also has n elements. Is there a ...
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0answers
33 views

Exploring the properties of the Srogenfrey Line

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
0
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1answer
29 views

how to find all values satisfing a function whose depends on another function?

How can I find all values $x>0$ such that $\int_0^x [t]^2 \, \mathrm{d}t=2(x-1)$? Does there exist an analytic solution to this problem? I mean a non iterative method to find these values, where $[...
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0answers
17 views

Is my parsing to symbolic logic of this statement correct?

Statement Prove that the natural number x is prime iff x > 1 and $\sqrt x$ there is no posi- tive integer greater than 1 and less than or equal to x that divides x. My parsing attempt into ...
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0answers
35 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| \...
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4answers
50 views

Proof that $A + 1 \leq e^A$ for all $A > 0$

I was reading a proof where at a certain point the prover uses the following inequality $$A + 1 \leq e^A$$ which in my opinion needs also a proof to be used around. I think I'm missing some ...
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3answers
26 views

How to prove a function from A to B

I have a question that says... THEOREM: The function $f: \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ given by $f(x) = ln(x)$ is onto. If you were going to prove this statement, what is the first ...
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2answers
38 views

How to disprove a theorem

I have a question that says, Explain how to disprove a theorem of the logical form "$\forall x \in A, P(x)$". Write the logical form of the statement you want to prove. So disprove a theorem, ...
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1answer
18 views

How to improve my proof and whether or not one condition in the statement is important in writing the proof?

A simple graph $G$ is connected iff for every partition of the vertices into two non-empty sets $X$ and $Y$, there is a vertex $x\in X$ and a vertex $y\in Y$ such that $xy$ is an edge of $G$. My ...
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2answers
22 views

Using the definition of the convergence of the sequence, how can I prove that this sequence converges to the limit?

I suppose what I am confused most on here is the algebraic method. $\lim_{x\to\infty} $ $ 2n^2/(n^3 + 3) $ = 0 I have set it up so far: Let $ \epsilon > 0 $ be given. I set up the equation ...
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1answer
20 views

Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.

As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations. ...
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1answer
48 views

Trying to Understand How to write Proofs

I am trying to study for a proofs final, and I'm really struggling with writing proofs. Does anyone have any suggestions that might help me to write proofs when given a theorem? I know there are ...
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3answers
42 views

What Proof Strategy to use

I have this theorem(see below) that I am trying to prove. However, I am struggling with how to get started; I don't understand what which proof strategy to use like proof by contradiction, if P then Q,...
0
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1answer
36 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that $\...
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2answers
28 views

Is $h(x)=\left(x^2; 1_{[0,\infty)}(x)\right)$ an injective function?

Let h defined by : $$\begin{align} h \ \colon\ \mathbb{R} & \to \mathbb{R}^{2} \\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ and $$\begin{align} f \ \colon\ \mathbb{R} &...
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2answers
25 views

Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph

Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
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0answers
49 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
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3answers
23 views

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$ Note the definition of closure I am using is one in Munkres: $x \in \overline A \iff \text{ for every open ...
3
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2answers
91 views

How can I prove this equation holds?

As the final part of a big proof I got for uni homework: (It is an extra question, may be unsolvable) $$k^n<\sum_{i=0}^n\binom{n}ik^{n-i}(2^i-1)$$ My idea is to develop the right side into an $(x+...
0
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1answer
31 views

$\mathbb{Z}\setminus U$ is open, where U is a basic open set of $\mathcal{B}$, the set of all arithmetic progressions

Let $m, b \in \mathbb Z$ with $m \neq 0$, and $U$ is of the form $Z(m, b) = \{ mx + b \mid x \in \mathbb Z \}$ I'm not sure how to show $\mathbb{Z}\setminus U$ is open, I was thinking to expressing $...
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0answers
36 views

Prof claimed that $\arcsin x + \arccos x$ does not always round to 90º under the same number of significant figures.

The core problem was simple: Determine the interior angles of triangle ABC given side $a = 12.34cm$ and hypotenuse $c = 35.32cm$. Simple. Clearly, the solution is: $\sin A = \frac{12.34cm}{35....
3
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2answers
38 views

Could anybody please check my proof about connected graph?

I have written a proof of the following statement but not sure whether it is correct or not. Let $G$ be a connected graph and each vertex has even degree. Show that if we remove ANY edge of the graph ...
2
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1answer
44 views

How to prove $\lim_{x \to \infty} \dfrac{x^a}{x^b} = 0$ for all nonnegative constants $a < b$.

I'm working my way through Mathematics for Computer Science at MIT OCW, and there is a lemma in the text that I am trying to prove and I've gotten stuck. $$\lim_{x \to \infty} \dfrac{x^a}{x^b} = 0$$ ...