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Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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1answer
22 views

ODE $f:\text{<0 if $tx > 0$},\text{>0 if $tx<0$}$; show that $x(t)\equiv0$ is the only solution to $\dot{x}=f(t,x)\hspace{0,3cm}, x(0)=0$

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ continuous fulfilling $$ \begin{cases} f(t,x)<0, & \text{if $xt>0$}.\\ f(t,x)>0, & \text{if $xt<0$}. \end{cases} $$ Show ...
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3answers
42 views

Autonomous ODE $\dot{x}=f(x)$: $\lim_{t\rightarrow\infty}x(t)=x^*\Rightarrow f(x^*)=0$

Let $x : [0,\infty) \to \mathbb{R}^d$ be a solution for the autonomous ODE $$\dot{x} = f(x)$$ where $f : \mathbb{R}^d \to \mathbb{R}^d$ is a Lipschitz continuous vector field. We know that $$\lim\...
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3answers
66 views

Can “and” be Substituted with “+” in proofs

Notes: Considering two limit were given $$\mathop {\lim }\limits_{x \to a} f\left( x \right) = L \hspace{0.1in} and \hspace{0.1in} \mathop {\lim }\limits_{x \to a} g\left( x \right) = M$$ means ...
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1answer
72 views

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ [duplicate]

I want to prove that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ There is a solution of the sum provided in my text book. There the expansion of $(1+\frac{1}{n})^n$ is like below: $(1+\frac{1}{n})...
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0answers
15 views

Find the equivalent of a sequence [duplicate]

let $u_n$ such as $u_0 > 0$ and $u_{n+1}=u_n+1/u_n$ Show that $u_n \sim \sqrt{2n}$
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1answer
48 views

Rigorous proof of a linear algebra theorem

I do seek a formal proof for the following statement. Let $V$ be a vector space such that $dimV=n$, let $S⊂V$ and let $v_1,...,v_r$ be a basis for $S$, then $S^⊥$ has $n-r$ linearly independepent ...
2
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3answers
60 views

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
2
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1answer
48 views

Example for Hilbert quote

Hilbert famously said The art of doing mathematics consists in finding that special case which contains all the germs of generality. Can you give an example of a situation in mathematics where ...
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0answers
30 views

Help: Hamming Code - Number of Parity Bits [PROOF]

I need to prove the following for the 1-bit correction hamming code by using equivalent transformations. $2^m (m+r+1) \leq 2^{m+r}$ => $r \geq 1 + \lfloor log2(m)\rfloor$ I have an Alphabet of size ...
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3answers
25 views

Prove that all finite sets are countable [duplicate]

How can I prove that all finite sets are countable? A set $S$ is countable if there exists an injective mapping $f:\mathbb{N}\rightarrow S$. So, shall I prove the above by contradiction? I assume ...
0
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0answers
14 views

Riemann integrals property proof [duplicate]

Prove that every function $f:[a,b]\rightarrow\mathbb{R}$ which is bounded and continuous everywhere but not in the finite number of points $x_1,...,x_n\in(a,b)$ has Riemann integral.
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2answers
24 views

Prove that all subsets of countable sets are countable

This is basically a problem of my assignment, it says... A set is said to be countable if it is either finite or there is an enumeration(list) of the set. Then prove that All subsets of countable ...
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0answers
13 views

The smoothing property of Bernstein polynomial

The smoothing property of Bernstein Polynomial, proved by Kelisky and Rivilin in 1967, $$ \lim_{k\rightarrow\infty}B\;^{(k)}\left(f;x\right)=\left(1-x\right)f\left(0\right)+xf\left(1\right) $$ can be ...
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2answers
28 views

Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

I'm studying for a qualifying exam in algebra, and my abstract algebra skills are quite rusty. I'm attempting to solve the following problem: Suppose that $\Phi:G\rightarrow\mathbb{Z}$ is a ...
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2answers
52 views

Proof of Binomial-coefficients sum [duplicate]

How could I show by induction that this sum is true? $${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+...+{n \choose n}^2 = {2n \choose n}$$ All help is appreciated!
5
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5answers
44 views

Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the Supremum of the above set? I consider the function $f(x)= x^{\frac{1}{x}}$, and show that $f(x)$ is maximum when $x=e$. ...
0
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1answer
19 views

Going from local to global definition in proving sets are manifolds

I'm new to manifolds and in exercise 4.2.1 from J.J. Duistermaat's Multidimensional Real Analysis I, I have to prove that $V = \{x\in\mathbb{R}^2 : x_2=x_1^3\}$ is a $C^{\infty}$ manifold. The ...
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2answers
34 views

Proving divisibility of integers [on hold]

Given integers $x$ and $y$ and a prime number $k>3$. It turned out that $x + y$ and $x^2 + y^2$ are simultaneously divisible by $k$. Prove that $x^2 + y^2$ is divisible by $k^2$?
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0answers
50 views

Countability and Proof

An infinite set is $\textbf{countable}$ if There exists a bijective function from the naturals to set. Want to prove: Let A be an infinite set. A is countable $\iff$ $A=$ $\{$ $a_1,a_2......$ $\}$, ...
4
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2answers
17 views

Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
0
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1answer
23 views

Proof by contradiction on if and only if statements

Suppose I want to prove a general statement like 'A is true if and only if B is true' If I assumed B is untrue and showed that subsequently A is untrue, which direction am I actually proving? I guess ...
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2answers
44 views

What's it called when, in a proof, we define a new variable/function in terms of two existing ones, in order to make it easier to write or follow?

Say, for instance, we're trying to prove a theorem involving two defined functions $f$ and $g$, and $x \in \mathbb{R}$. Within this proof, we have to deal with the sum $f(x) + g(x)$ multiple times. ...
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2answers
26 views

Proving goal having the form $P \lor Q$, is it redundant to separate into two cases?

In Velleman's How to Prove It, the strategy given for proving goal of the form $P \lor Q$ goes like this: If $P$ is true, then clearly $P \lor Q$ is true. Now suppose $P$ is false. ...
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4answers
50 views

Show set equality.

I know that to show set equality you must show that the two sets are subsets of each other. I'm having trouble showing that S\T is a subset of (S U T) given the assumption that T is a subset of S. I ...
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5answers
31 views

what's the best way to prove the equivalences of such formulas?

I want to prove the following: $$2^n+2^{n-1}+...+2^1 + 1 = 2^{n+1}-1$$ The only Method that I know of is proof-by-induction but is this the best way to prove the equivalences of such formulas?
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1answer
36 views

Proof that any integer $z>1$ can be written as $2x+y$, where $x>y$

Imagine a multiple choice questionnaire with 3 choices $a, b,$ and $c$. At the end the sums of each choice are tallied. It seems it's always possible to have a tie for first, as long as the total ...
0
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1answer
29 views

Prove these differently written de morgan laws

We define $$\overline{\bigcup_{p\in P} Sp} =\bigcap_{p\in P} \overline{Sp}$$ and $$\overline{\bigcap_{p\in P} Sp} =\bigcup_{p\in P} \overline{Sp}$$ Which are just another way to write de morgans laws....
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1answer
63 views

Showing that two numbers are the same percent different from their average.

More specifically, consider two real numbers $a,b>0$, and their average $r=\frac{a+b}{2}$. It is the case that $a=r*x$ and $b=r*y$ where $\vert 1-x\vert =\vert 1-y\vert$. For example, let $a=5$ ...
2
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1answer
57 views

a < b if and only if a++ ≤ b.

I have to prove that a < b if and only if a++ ≤ b. I am using the book analysis 1 by Terence Tao, which unfortunately has no section for solutions of exercises. both a and b are natural numbers, ...
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2answers
68 views

Proving $2\cosh 2x+ \sinh x = 5$ [closed]

I have been sitting on this question for quite some time and I haven't been able to prove this identity. Please anybody who can help me here. I am new to hyperbolics. $$2\cosh 2x+ \sinh x = 5$$ I ...
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0answers
19 views

Prove infinite markov reward process converges

The following question is obtained from Stanford CS234 Lecture 2 notes, Excercise 3.7 Let $r_i$ denote the reward obtained from transition $s_i\rightarrow s_{i+1}$. Furthermore, the return $G_t$ of a ...
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3answers
33 views

Prove that $0< \frac{1}{2^{m}} <y$

If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$ I think I have to use Archimedean property to prove it. The Archimedean property ...
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0answers
22 views

Proving following theorem, Grönwall

Suppose $y:[t_0,+\infty) \to [0,+\infty)$, $t_0 \in \mathbb{R}$, is a non-negative continuous function and $u:[t_0,+\infty)\to\mathbb{R}$ is a non-decreasing continuous function. Let $L\in\mathbb{R}^{+...
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1answer
19 views

Show that the set of LES-solutions form a sub-vector space of $\mathbb{R^n}$ exactly when $b_i = 0$

The linear system of equations is given: \begin{align} a_{11}x_1+\dots& +a_{1n}x_n=b_1\\ &\vdots\\ a_{m1}x_1+\dots&+a_{mn}x_n=b_m \end{align} Show that the set of the given linear ...
1
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1answer
42 views

Exact ODE: show that $y$ is a solution iff it is in a level set of $F$

A Differential equation of the form $p(x,y(x))\dot{y}(x)+q(x,y(x))=0\hspace{1cm}(1.1)$ is called exact, if there is a differentiable function $\mathbb{R}^2\mapsto\mathbb{R}$ such that ...
1
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1answer
35 views

Proving $\frac{1+\csc^2A\tan^2C}{1+\csc^2B\tan^2C}=\frac{1+\cot^2A\sin^2C}{1+\cot^2B\sin^2C}$

Prove $$\frac{1+\csc^2A\tan^2C}{1+\csc^2B\tan^2C}=\frac{1+\cot^2A\sin^2C}{1+\cot^2B\sin^2C}$$ I chose to manipulate the left hand side of the equation, by firstly replacing $\cot^2A$ with $\csc^2A-...
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0answers
52 views

Formulating ordinary problems mathematically in order to solve them

I've been thinking about an ordinary problem for which there doesn't seem to exist a solution given its constraints. I was wondering how would one go about formulating the problem mathematically such ...
4
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3answers
111 views

linearly independent solution to second order ODE.

Let $y(t)$ be a nontrivial solution for the second order differential equation $\ddot{x}+a(t)\dot{x}+b(t)x=0$ to determine a solution that is linearly independent from $y$ we set $z(t)=y(t)v(...
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1answer
45 views

Does $n^\prime\ne n^{\prime\prime}$ require proof by contradiction? $n^\prime$ is the successor of $n$.

This is the statement of Peano's axioms I will assume for this discussion: $1$ is a number. To every number $n$ there corresponds exactly one number $n^\prime.$ $n^\prime=m^\prime\implies n=m.$ $n^\...
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1answer
38 views

Two Proofs for Open Sets and Metric Subspaces

I have two proofs for the following theorem: Let $(S, d)$ be a metric subspace of $(M, d)$, and let $X$ be a subset of $S$. Then $X$ is open in $S$ if and only if $X = A \cap S$ for some set $A$ ...
1
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2answers
45 views

Proving an inequality involving absolute values

How can I prove the inequality $\left|x\right|+\left|y\right|+\left|z\right|\le\left|x+y-z\right|+\left|y+z-x\right|+\left|z+x-y\right|$ for all $x, y, z$ being real number. Can I prove this by ...
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2answers
27 views

Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$

Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$ Working:$\frac{x+y}2=0$ or, $x=-y$ I plotted this. Plotting $|x| + |y| = 1$, I got ...
0
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1answer
27 views

Show that with the exception of (3,5) all twin primes are of the form (6k -1 , 6k +1). [duplicate]

Show that with the exception of $(3,5)$ all twin primes are of the form (6k -1, 6k +1) for some k. My question is: Why we have the number 6 beside k ? Also any hint for the proof is appreciated.
1
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1answer
19 views

Prove that all terms in an arithmetical equations are equals with border conditions

Would it be possible to prove that there is an equation that includes a number N of unknown numbers that are all equal, between 0 and 1 and whose sum is equal to 1 ? And to find this equation ? I ...
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0answers
26 views

Question about how to approach existence in proofs

I am working through some problems in Axler's Linear Algebra Done Right textbook, and I noticed that I haven't really developed an intuitive feel for how to approach existence in the proofs. The idea ...
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1answer
23 views

Let (x+a) be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$ [closed]

Let $(x+a)$ be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$.
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0answers
26 views

Basic negation of a statement

Statement is $\exists I = (x_o - \frac{1}{n}, x_o + \frac{1}{n}), n \in \mathbb N$, s.t $f(x) > 0 $ $\forall x \in I$ When negating the part after "such that", would it be $f(x) \leq 0$ $\forall ...
2
votes
5answers
87 views

Does the non-commutativity of quaternions follow directly from $\rm i^2=j^2=k^2=ijk=-1$?

All of quaternions are, from what I understand, defined simply by $$\newcommand{\i}{\mathrm{i}} \newcommand{\j}{\mathrm{j}} \newcommand{\k}{\mathrm{k}} \i^2=\j^2=\k^2=\i\j\k=-1$$ It is known that ...
0
votes
1answer
62 views

needing help for proofing $\frac{de^x}{dx}=e^x$

could anybody explain why do we proof $\frac{de^x}{dx}=e^x$ in this way "we know $y=e^x=f(x)$ and $(f(y)^-1)' =\frac{1}{f'(x)}$ ,so $\frac{dy}{dx}=\frac{de^x}{dx}=\frac{dln^-1}{dx}=\frac{df(x)^-1}{dx}...
2
votes
0answers
31 views

Prove that if $x$ is element of the group G then $H = \lbrace x^n : n \in Z\rbrace$ is a subgroup of $G$.

I am looking to prove Dummit and Foote Chapter $1$ problem $27$. The other proofs I have seen are longer so I feel like there is something I'm missing or my proofs aren't as clear as they should be. ...