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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36 views

Prove that the sequence $\{n^4\}_n$ diverges to infinity.

My question is how would you go about proving this problem? Proofs are not my strong suit as I prefer solving for a specific answer. Anyway I read somewhere that you would prove the reciprocal ...
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15 views

Is there a way to add a node to a digraph without adding a new topological sort?

I have a digraph with $n$ nodes. I want to add a new node to the graph, but I don't necessarily want to add a new edge to connect this node. In essence I want the graph to remain acyclic. Can I add ...
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1answer
26 views

Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f(c+\frac{1}{2})$.

Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f\left(c+\frac{1}{2}\right)$. So, I know I'm supposed to use the ...
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1answer
18 views

Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem.

Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem. Uhhh I have no idea where to even start with this. Anything to give ...
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1answer
28 views

Show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution

As it says in the title, I would like to show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution. However the only thing I know for the ...
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1answer
72 views

The set of all rational points in the plane is a countable set

From Kolmogorov's Introductory Real Analysis. I am doing some self-study and would like some feedback on whether my proof is correct. I am using that the set of rational numbers is countable as given, ...
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0answers
16 views

Proving a summation formula using the general Leibniz rule

I am trying to prove the following relations: $$ \partial^{N-2}(f^{N-1}g) =\sum_{n+m=N-2}\frac{(N-2)!}{n!\,(m+1)!}\,\big[\partial^{n}(f^{n}g)\big]\,(\partial^{m}f^{m+1}), \qquad N\geq2, $$ and $$ \...
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1answer
25 views

Prove energy preservation of implicit midpoint method.

I am using the Runge Kutta implicit midpoint method $$m_{n+1}=m_n + \frac{h}{2}(\frac{m_n + m_{n+1}}{2} \times (T^{-1}\frac{m_n + m_{n+1}}{2}).$$ To solve a free rigid body problem, where $T$ is the ...
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Finding analytic continuatiuon of a branch

I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases. The ...
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0answers
44 views

Positive function not vanishing in a neighbourhood

If a positive function in $C[-1,1]$ does not vanish in any neighbourhood of $-1$, then there exists $\epsilon>0$ such that $f(x)>0$ for all $x\in (-1,-1+\epsilon)$. it has to be strictly ...
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0answers
37 views

Show infinitude of primes by number of product of factors.

Question from book on Introductory Number theory, by Andre Weil, chapter 4, problem IV.6. If $n, a, b,..., c$ are integers $>1$, then the number of distinct integers of the form $a^{\alpha}b^{\beta}...
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1answer
34 views

If a is non zero non unit in a PID then a has at least one irreducible divisor?

Sources for the proof of this claim: If $a$ is non zero non unit in a PID then $a$ has at least one irreducible divisor. Definition of irreducible element: a non zero non unit element $p$ is ...
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3answers
37 views

Prove that this series is absolutely convergent.

For this question, I want to be able to either prove or disprove that If $a_n \not= 0 \forall n \geq 2$ and $\sum_{n=2}^{\infty}a_n$ is absolutely convergent, then $\sum_{n=2}^{\infty}\frac{na_n}{n-2}...
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1answer
100 views

Find all pair of primes $(p,q)$ such that both $p^2+q^3$ and $p^3+q^2$ are perfect squares.

Let $p^2+q^3=a^2$ and $p^3+q^2=b^2$. Let's suppose $ p \neq q$. When one of $p,q$ equals $2$, it yields system of equations with no solution, so $p,q \geq 3$. Since any two primes numbers are coprime,...
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1answer
25 views

How to prove this is a partial order?

A relation is defined by: $x\leq y$ if and only if there exists $𝑘\in \mathbb{N}$ such that $y= x+5k$. Prove that $\leq$ is a partial order. I have no idea how to do this question. I've tried my best ...
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0answers
20 views

How can I prove that an invented operation works with negative numbers? [duplicate]

Just out of curiosity I tried making a new operation to see its properties. It is like a variation of addition that works like this: a ¨ b = a + b - 1 For example, I found that it has the commutative ...
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2answers
50 views

Square free proof on if $b|a^2$ then $b|a$

We say that b is square-free if $b$ can be written as the product of distinct prime factors. That is, $b = p_1 \dotsm p_n$ for $p_1$ does not equal $p_2$ which does not equal $\dotsc p_n$ Let $a, b \...
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1answer
27 views

Let $P$ be an idempotent linear operator on V. Then if $\text{null}(P)\subseteq(\text{Im}(P))^\perp$, $P$ is an orthogonal projection.

I'm trying to understand why the statement above is true for a finite dimensional inner product space V. After some research I found that as long as $P$ is a linear operator on V and is idempotent, ...
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0answers
79 views

Formalizing intuition on real analysis

Let $f:\mathbb R\to \mathbb R_+$ be a function such that: $\int f(u)du=1$; It is differentiable on its support $[0,\infty)$; It is bounded (there is $c_1>0: \lvert f\rvert \leq c_1$); There is $v&...
2
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1answer
36 views

Proof a mapping is convergent by Cauchy

I want to prove the following is convergent by Cauchy, i.e. for any $\epsilon >0$, there is some $N$ such that for every $m \ge N$ and every $n\geq 0$, $|x_{m+n} - x_m| < \epsilon$. Given: $g$ ...
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0answers
45 views

How can I prove these affirmations?

Let $n$ be a natural number. Consider the number $X_n = 111\ldots111$ made by $n$ repeated digits of $1$. a) Prove that, for a prime $X_n$, $n$ is prime. b) Prove that, if $X_n$ isn't prime, there ...
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1answer
17 views

peano arithmetic proof in fitch

I've been tasked with proving that any natural number times the successor of zero is equal with that natural number. I've been trying to solve this problem using induction in the Fitch proof system, ...
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0answers
37 views

Proof of Absolute Convergence of Power Series

Any tip on how to prove the absolute convergence. I tried to assign a series function for An and work my way through that, but I was wondering if there is a more efficient method? The picture got cut ...
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0answers
41 views

Baby Rudin 2.34 proof style and logical leaps

There are already maybe a dozen questions about this proof on this site. This question contains the problem statement; I'm interested in the conclusion that there are finitely many points $q_i$ in $K$ ...
2
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0answers
42 views

Existence of $\alpha : K ⊗_F V \stackrel\sim\to K ⊗_F W$ implies the existence of $\beta : V \stackrel\sim\to W$

Let $V$ and $W$ be finite-dimensional vector spaces over a field $F$, let $\varphi : V \to V$ and $\psi : W \to W$ be $F$-linear endomorphisms, and let $K/F$ be a field extension. Denote by $\varphi_K$...
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1answer
34 views

Proof of a convergence of a given probability

I am studying probability theory and convergence is something that is very new to me. So I am trying to solve the mentioned problem and any guidance is appreciated. Let $X_n \xrightarrow d $ $X$, $f(n)...
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1answer
33 views

Prove or disprove that this is a polyhedron.

Is $$\{(x,y) \in \mathbb{R}^2 ; x^2+y^2 \leq 1, x -5y=0\},$$ a polyhedron? I know that the unit disk, $$\{(x,y) \in \mathbb{R}^2 ; x^2+y^2 \leq 1, x \geq 0, y \geq 0\},$$ is not a polyhedron. How ...
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1answer
38 views

Prove that: $\mathbb{P}(A \backslash B) = \mathbb{P}(A) - \mathbb{P}(B), B \subseteq A \subseteq \Omega \land \mathbb{P}(B) \leq \mathbb{P}(A)$

Prove that: $$\mathbb{P}(A \backslash B) = \mathbb{P}(A) - \mathbb{P}(B), B \subseteq A \subseteq \Omega \land \mathbb{P}(B) \leq \mathbb{P}(A)$$ Additionally, we know that for all $A, B \subseteq \...
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1answer
39 views

How to prove that for all odd $ n \in \mathbb{N} $ can be displayed as the difference of two square numbers?

I need guidance / correction for my proof. It's a little bit longer, but we really have to consider everything. If you find some issues / mistakes or have suggestions to improve it, please let me know!...
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1answer
37 views

How to go about proving the sequence is Cauchy

The statement of the question is: Suppose $f$ maps the open interval $E$ into itself, $0 < b < 1$, $f$ has property $X(b)$ (that property is Lipschitz continuous), and $x_0 \in E$ Prove that the ...
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0answers
33 views

mathematical induction in $\mathbb{Z}$ Terence Tao [closed]

I am self studying Terence Tao book. One of the exercises asked us to show the following. Show that the principle of induction (Axiom 2.5) doesn't apply directly to integers. More precisely, give an ...
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1answer
29 views

Show case when minimal polynomial coincides with its characteristic polynomial

As is introduced in the title, I'm stuck on the following problem: Considering a linear endomorphism $φ$ of an $n$-dimensional vector space $V$ having $n$ pairwise distinct eigenvalues, I would like ...
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0answers
29 views

Spectrum of $\phi\otimes\psi$.

I'm wondering if somebody can help me to prove this statement : Suppose that $\phi$ and $\psi$ have characteristic polynomials completely decomposable, then the spectrum of $\phi \otimes \psi$ is {$\...
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1answer
42 views

What proof type is best for math competitions? [closed]

What proof type is best for essay mathematics competitions like USAJMO, USAMO, and IMO? How can you get used to writing proofs?
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0answers
24 views

Prove that $(\lambda^2,v)$ is an eigenpair

Let the characteristic polynomial of a 3x3 matrix be $-t^3+4t^2-6$. And $(\lambda, v)$ is an eigenpair. Prove that $(\lambda^2, v)$ is an eigenpair of $A^2$ and $(\lambda^3,v)$ is an eigenpair of $A^3$...
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0answers
32 views

Is this statement correct? How would I go about proving it?

There is an optimization for checking if a number is prime via "brute-force". It states that, given a positive integer $x$, one simply needs to check that all integers less than or equal to $...
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1answer
31 views

Proof of Graph Subdivision

Prove that every simple graph of order $n ≥ 4$ and size at least $2n − 2$ contains a subdivision of $K_4$.
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1answer
78 views

Problem on Graph Coloring

A planar map is called non-degenerate if all vertices have degree $3$ that is borders of only $3$ countries meet at a point. Suppose that in a non-degenerate planar map, all faces have an even number ...
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1answer
16 views

Proving that a Jordan basis is not uniquely determined. [duplicate]

It seems that a Jordan basis, for, say, an operator $φ:V→V$ with $V$ a vector space can have several Jordan bases. However I don't see how this is true and I didn't achieve to prove it. Indeed I tried ...
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1answer
16 views

Unique four-regular, simple planar graph such that every face is bounded by three edges

A question on my graph theory exam asked us to find how many $4$-regular, simple planar graphs there are up to isomorphism such that every face, including the outer face, is bounded by three edges. I ...
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1answer
29 views

Show that $f_n(x)=n(f(x+\frac{1}{n})-f(x))\to f'$ uniformly (with $f \in C^2$ and $f''$ bounded)

Let $f:\mathbb{R}\to \mathbb{R}$ such that $f \in C^2$ and $f''$ is bounded. Show that the sequence of functions defined by $f_n(x)=n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)$ converges uniformly ...
3
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2answers
60 views

Proving $\Gamma$ function is a group

I saw this question yesterday in stack-exchange ,but it was suddenly deleted by OP. Fortunately , i could find the original question thanks to the source given by OP. According to OP , the function $\...
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1answer
27 views

$A_i \subset B_i$ and $\bigcap A_i$ finite while $\bigcap B_i$ infinite, then $\bigcap B_i \setminus A_i$ is also infinite.

Proposition. If $|\bigcap_{i \in I} A_i | \lt \infty, \ B_i \supset A_i, \ \forall i \in I, \ $ and $|\bigcap_{i \in I} B_i| = \infty$, then $|\bigcap_{i \in I} (B_i \setminus A_i)| = \infty$. Proof. ?...
2
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1answer
30 views

Validation of a result (logic)

Suppose that the nonnegative function $f$ is such that for any $v>1$ there are $C,L>0$ so that $f(x)\leq Cx^{-v}$ for all $x>L$. It is clear that $\lim_{x\to\infty}f(x)=0$. Is it true that $f(...
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1answer
38 views

Proving a metric space to be separable if every infinite set in the metric space has a limit point in it.

Let $X$ be a metric space such that every infinite set in it has a limit point. I want to prove that $X$ is separable that is $X$ has a countable dense subset. $X$ may either be countable or ...
2
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1answer
36 views

contraction mapping and convergence proof

I am having difficulties arranging and concluding the proof... Suppose $f$ maps the open interval $E$ into itself, $0 < b < 1$, $f$ has property $X(b)$ (that property is Lipschitz continuous), ...
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1answer
19 views

On an open interval prove that if $f$ is differentiable and |f'(x)| \leq C then $f$ is Lipschitz continuous

I'm having trouble understanding how to work this out, been working on it for hours to no avail...This is a question in my homework: Statement of the problem: suppose that E is an open interval. Prove ...
1
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1answer
58 views

Solving the functional equation $(f (x + y))^ 2 = f (x^ 2 ) + f (y ^2 )$ [closed]

Please I need help with understanding how to solve functional equations. I have a question I'd like to examine below. I have tried to learn the methods of proofing in mathematics (e.g. induction, ...
2
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0answers
49 views

Proof writing clarification.

Problem Let $X_n$ be a sequence of mixing random variables. If there is $a>1$ such that the strong mixing coefficient satisfies $$\alpha(s)\leq Cs^{-a}\quad (1)$$ ($C>0$ is a generic positive ...
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1answer
26 views

Use the definition of continuity to prove that $g$ is continuous at $x = 0$

Let $D$ be a subset of R containing $0$, and let $f : D → R$ be bounded on $D$ (i.e., $f(D)$ is a bounded subset of R). Define a new function $g : D → R$ by $g(x) = xf(x)$. (a) Use the definition of ...

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