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.
13,515
questions
-1
votes
2answers
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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, ...
0
votes
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
$$
\...
0
votes
1answer
25 views
Prove energy preservation of implicit midpoint method.
I am using the Runge Kutta implicit midpoint method
$$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 ...
0
votes
0answers
12 views
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 ...
0
votes
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 ...
0
votes
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}...
-2
votes
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 ...
0
votes
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}...
3
votes
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,...
0
votes
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 ...
1
vote
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 ...
0
votes
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 \...
1
vote
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, ...
0
votes
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
votes
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$ ...
0
votes
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 ...
0
votes
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, ...
0
votes
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 ...
1
vote
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
votes
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$...
2
votes
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)...
-2
votes
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\},$$ is not a polyhedron. How ...
0
votes
1answer
38 views
Prove that: $\mathbb{P}(A \backslash B) = \mathbb{P}(A) - \mathbb{P}(B), B \subseteq A \subseteq \Omega \land \mathbb{P}(B) \leq \mathbb{P}(A)$
Prove that: $$\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 \...
0
votes
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!...
1
vote
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 ...
-2
votes
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 ...
0
votes
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 ...
0
votes
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 {$\...
0
votes
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?
0
votes
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$...
0
votes
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 $...
0
votes
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$.
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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
votes
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 $\...
1
vote
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
votes
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(...
0
votes
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
votes
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), ...
0
votes
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
vote
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
votes
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 ...
0
votes
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 ...
