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

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2
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2answers
62 views

Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
2
votes
1answer
498 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
0
votes
2answers
65 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 ...
1
vote
1answer
26 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 ...
0
votes
1answer
35 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 : ...
0
votes
1answer
33 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 ...
2
votes
1answer
32 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 ...
3
votes
6answers
46 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 ...
1
vote
1answer
49 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
votes
1answer
44 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 ...
1
vote
3answers
34 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 ...
2
votes
1answer
52 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
votes
1answer
25 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 ...
1
vote
1answer
16 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 ...
-1
votes
2answers
27 views

How to approach Proofs? [on hold]

I'm currently in a CS course and a big part of it is proving proofs. I am struggling and want to become better. I have always struggled with proofs ever since I first encountered them in highschool - ...
0
votes
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 ...
5
votes
3answers
151 views

Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
0
votes
1answer
389 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
0
votes
0answers
23 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
votes
2answers
686 views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
1
vote
1answer
21 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 ...
0
votes
0answers
33 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}| ...
1
vote
0answers
21 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 ...
2
votes
2answers
34 views

Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
1
vote
1answer
17 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 ...
5
votes
2answers
544 views

How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
votes
1answer
24 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 ...
0
votes
0answers
15 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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. ...
1
vote
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, ...
0
votes
1answer
808 views

Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or ...
1
vote
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 ...
-1
votes
0answers
35 views

How to prove that $p$ divides $a^p -a$ for every integer $a$. [closed]

How to prove this Fermat's little theorem: $p$ divides $a^p -a$ for every integer $a$.
-4
votes
0answers
36 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
1
vote
2answers
18 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
29 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 ...
1
vote
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} ...
1
vote
2answers
811 views

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in A has its limit in A

Real analysis is a topic I'm unfamiliar with and I'm confused on how to write proofs on them. In order to prove that: A set $A \subseteq \mathbb{R}$ is closed (1) $\iff$ Every convergent sequence in ...
3
votes
2answers
37 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 ...
0
votes
0answers
45 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
2answers
151 views

Can some inequalities help to pin down an unique solution in a linear system of equations with infinite solutions?

I need to discuss the number of solutions of the following system of equations. Any help would be very appreciated. Consider the known parameters $a_1,...,a_4;d_1,d_2,d_3$ such that $0< a_i< ...
0
votes
2answers
34 views
3
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2answers
90 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 ...
0
votes
1answer
26 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 ...