For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

learn more… | top users | synonyms

1
vote
0answers
44 views

Working with the Mobius transformatios and linear algebra.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
0
votes
0answers
30 views

Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...
1
vote
0answers
16 views

countable zeros of a particular solution to some 2nd order differential equation

Consider the differential equation$: \ e^xx^2y''-e^xxy'+(x^2-1)y=0.$ Suppose $f:(-\infty,0) \to \mathbb{R}$ is such that $(1-x^2)f(x)=e^x(x^2f''(x)-xf'(x)), \forall x\in (-\infty,0).$ If $f$ is not ...
1
vote
2answers
33 views

Prove that arbitrary tetrahedron can be intersected with a plane so that in the intersection will be a rhombus

Prove that arbitrary tetrahedron can be intersected with a plane so that in the intersection will be a rhombus. My idea is to find $4$ coplanar points $A,B,C,D$ (vertices of rhombus) on the pyramid ...
2
votes
1answer
55 views

“Half-primitive root”?

I've made a topic related to this (but containing a different question) and it got no responses, so I was wondering if I've stumbled on something new or if it's obvious and I'm just not seeing it at ...
0
votes
1answer
27 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
1
vote
1answer
46 views

Inequalities - proof by induction

Proof by induction involving inequalities completely escapes me. I've encountered the following problem: For which non-negative integers n is $n^2 ≤ n!$? Prove your answer (by induction). So, ...
0
votes
1answer
21 views

inductively defined group statements

If A is a set, and $B_{1}, B_{2}\subseteq A$ subsets of $A$. Also, $f_{1}:A\rightarrow A$ and $f_{2}:A\rightarrow A$ we will mark: $F1={f1} , F2={f2}$ How do I prove the following: $X_{B1\cap ...
2
votes
1answer
34 views

When there is a proposition $(P\rightarrow Q)$, which row in the truth table of $\rightarrow $ should I use?

I solved one question in a book of analysis, and although I used an informal method to check it, I'd like to know more about what should be done. The question was the following: $A\subset X$ ...
2
votes
0answers
19 views

Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
1
vote
1answer
32 views

Poincare Inequality for 1-Dimensional Problem.

I am referring to the book Introduction to Functional Analysis to Boundary Value Problems and Finite Element by Daya Reddy (page ...
2
votes
1answer
22 views

Is $T:= \{g \in A_4|g^2 =(1)\}$ a subgroup of $A_4$?

Consider the subset $$T:= \{g \in A_4|g^2 =(1)\}$$ of the alternating group $A_4$ in four letters. Is T a subgroup of $A_4$? My Proof: Yes. If I am not wrong T is the Klein 4-group since only ...
0
votes
0answers
39 views

Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
-1
votes
1answer
30 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
-1
votes
0answers
41 views

Round table and division of numbers, need proof.

Let's assume that k-number of people are sited on a round table (k>=2). Each of them chooses a card with a number from 1 to n where n>=k. Each card has a different number (2 people can't pick a card ...
1
vote
2answers
86 views

Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$

Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$ for any $x\in\mathbb{R}\setminus{k\pi}$ where $k\in\mathbb{Z}$. I wrote $\sin5x$ as $5\cos^4x\sin{x}-10\cos^2 x\sin^3x+\sin^5x$ and ...
1
vote
3answers
111 views

Trivials are not easy to prove.

If $x,y\in \mathbb{R}$ and $x\neq y$, then show that there are neighborhoods $N_x$ of $x$ and $N_y$ of $y$ shuch that $N_x \cap N_y=\emptyset.$ I know the result is trivial but trivial things are ...
0
votes
1answer
38 views

Proof: $\sum\limits_{n=1}^\infty \mathbb E(|X_n|)< \infty \Rightarrow \sum\limits_{n=1}^\infty X_n$ converges almost surely

I was reading this as a Lemma, however my book doesn't provide proof of it: Let $X_1,X_2,...$ be a sequence of random variables, then the expression in the title is true. I'm trying to ...
2
votes
3answers
56 views

Induction Proofs - Mathematics

How do I show by mathematical induction that $2$ divides $n^2 - n$ for all $n$ belonging to the set of Natural Numbers
2
votes
3answers
59 views

Combinatorial Argument with Natural Numbers

Give a combinatorial argument to show that all natural numbers c(n,k) = c(n,m) where c stands for combination.
1
vote
0answers
36 views

Prove that a set with n elements in union with an element not in the set has n+1 elements

Suppose $A$ has $n$ elements and suppose $a \notin A$, prove that $A \cup \{a\}$ has $n+1$ elements. I am pretty sure that I am supposed to use something with one to one and onto functions. It all ...
0
votes
2answers
61 views

Spivak GENERAL limit law proof

Suppose $f(x) \le g(x)$ for all real $x$ Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$ Let limit for $f(x)$ be denoted by $L$ Let limit for $g(x)$ be denoted by $M$. ...
0
votes
2answers
20 views

Function bijective proving.

Let $\mathbb{C}$ be the set of all complex number. $z\in \mathbb{C}$ Given a function $$ f : \mathbb{C} \to \mathbb{C} $$ $$f(z) = (1+2i)z+5i$$ Prove that it is bijective. First, prove ...
9
votes
1answer
120 views

Functions that satisfy $f(x+y)=f(x)f(y)$ and $f(1)=e$

My real analysis professor mentioned in passing that there exist functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy all of the following conditions for all $a,b \in \mathbb{R}$: $$f(1)=e$$ ...
0
votes
2answers
85 views

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and $N$ since M ...
0
votes
1answer
34 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
1
vote
1answer
25 views

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
1
vote
3answers
39 views

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$ Should I prove this by induction? If so, how should I go about it?
5
votes
4answers
69 views

Prove that $ \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+\cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ for $n\in \mathbb N$

I want to prove that if $n \in \mathbb N$ then $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+ \cdots+ \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}.$$ I think I am stuck on two fronts. First, I don't know ...
1
vote
1answer
106 views

The fix points of the Möbius transformations are the eigenspace of a certain matrix.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
1
vote
3answers
41 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
votes
1answer
24 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
0
votes
1answer
26 views

Proving: If a function is bounded, then the fuction's limit is bounded.

The question I have to answer is the following: Let I be an open interval that contains the point c and suppose that f is a function that is defined on I except possibly at the point c. If $m \le ...
3
votes
2answers
120 views

Can the choice of epsilon be arbitrary in epsilon-delta proofs?

I've been reading Spivak's chapter on limits and something that I don't feel I understand entirely is how the epsilon is decided upon. It makes sense to me in the context of $\,|f(x)-L|<\epsilon$ ...
2
votes
0answers
31 views

Proof without using the proof of contradiction

By using the proof by contradiction I can determine that the root of a prime number is irrational. But how can I proof this by using the rational roots test to find rational factors of $x^n - p$. How ...
0
votes
1answer
40 views

Strong induction on a sequence, proving two functions are equal?

Excuse the poor title, but my understanding is still a little fuzzy. Admins feel free to change it Here is the question from the book. suppose that $f_{0}, f_{1}, f_{2}...$ is a sequence defined ...
2
votes
4answers
43 views

Proving binomial coefficients identity [duplicate]

Let $n$ and $r$ be positive integers with $n \ge r$. Prove that: $$\begin{pmatrix}r\\r\end{pmatrix} + \begin{pmatrix}r+1\\r\end{pmatrix} + \dots + \begin{pmatrix}n\\r\end{pmatrix} ...
0
votes
0answers
44 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
1
vote
1answer
35 views

Finding all z (complex) that satisfies an equation

I'm having a little trouble with this problem. It's asking to find all $z\in\mathbb C$ that satisfy $z^3 = -2(1+i\sqrt{3})\overline z$, and to keep the answers in standard form. I tried expanding ...
0
votes
1answer
19 views

prove characterstic polynomial of $2\times 2$ matrix is $C_{A}(x)=x^2-(\lambda_{1}+ \lambda_{2})x+\lambda_{1} \lambda_{2}$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $\lambda_{1}, \lambda_{2}$ not necessarily distinct, be the eigenvalues of A. Show that $$ ...
1
vote
1answer
38 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
1
vote
0answers
43 views

To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
0
votes
2answers
49 views

How do I prove this using proof by contradiction

There is a set a set $S$ of numbers. i.e. $(s_1, s_2, s_3, s_4, s_5, ..., s_n)$. The average of the numbers in the set is $N$. How do I prove that at least one of the numbers in the set is greater ...
0
votes
0answers
27 views

Poles of analytic functions are isolated

Can the set of poles of an analytic function $f:G\rightarrow \mathbb{C}$ contain a limit point? I know that the answer is no for open $G$, but after thinking more I have become paranoid about ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
4
votes
1answer
105 views

Product identity for $n^n$

I came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k ...
0
votes
3answers
45 views

Divisibility induction proof - question about fractions

I have a question about the example of divisibilty induction proof. Here's the problem [the expression must be divisible by 8]: $5^{n+1} + 2*3^n + 1 = 8*k$ I know that probably I have to proceed ...
0
votes
1answer
18 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...
0
votes
1answer
21 views

Prove the length of a linear combination of an orthonormal basis

Let $B=\{u_1,u_2,...,u_p\}$ where $B$ is an orthonormal basis for a subspace $W$. Let $v$ be any vector in $W$, where $v=a_1u_1+a_2u_2+...+a_pu_p$. prove that $$||v||^2=a_1^2+a_2^2+...+a_p^2$$ So ...
4
votes
0answers
56 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...