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

3
votes
3answers
52 views

If a set is countable and infinite, there is a bijection between the set and $\mathbb{N}$

I'm trying to show that if a set $S$ is infinite and countable then there is a bijection $\varphi : S\to \mathbb{N}$. Since $S$ is countable, we know that there is an injection $f: S\to \mathbb{N}$. ...
1
vote
1answer
8 views

Binomial Distribution formula

If $X\sim B(n,p)$, show that $P(X=r+1)=P(X=r) \cdot \frac{p(n-r)}{q(r+1)}$ for $r=0,1,...,n-1$ My attempt, $P(X=r+1)={_n}C_{r+1}(p)^{r+1}(1-p)^{n-(r+1)}$ How to proceed then?
0
votes
2answers
48 views

Proof for $V \cong V^{**}$

Theorem: Let $V$ be an vector space. Then the dual space of $V$'s dual space is canonically isomorphic to $V$. I am able to prove that $V$ is a subspace of $V^{**}$, the map ...
0
votes
0answers
15 views

Proving inequalities with the Archmedian property

I have to determine how large n∈N must be to ensure that (1/n)<ε is satisfied and use the Archimedean property to establish that such n exists. I know that the Archimedean property is ∀ε>0 and ...
2
votes
1answer
35 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
1
vote
1answer
26 views

Proving a given set is a submanifold

Let $S \subseteq \mathbb R^n$. I have been faced with showing that $S$ is a submanifold and I have some ideas but I want to get the complete picture. (Main) Question 1: What methods are there to ...
0
votes
0answers
16 views

Absolute Value Inequality of Differences

I'm hoping someone could give insight as to how I can improve my organization, and/or thought process. Show that $|a-b| \lt c$ if and only if $b -c \lt a \lt b + c$. By the statement $b - c \lt a ...
3
votes
4answers
105 views

Proving $\frac{n^n}{3^n} < n!$ for $n\ge6$ by induction

How would I prove this using mathematical induction: $\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$. Here is what I have tried: $\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$ Base case: ...
0
votes
0answers
30 views

Absolute Value Inequality Proof

I realize this is almost identical to another question I posted, but I wanted to ask what the distinction between the two is -- comprehension-wise (other than the $\lt$ vs. $\le$). Show that $|b| ...
0
votes
0answers
44 views

A question regarding a theorem of Erdos and Hajnal

Consider the following theorem of Erdos and Hajnal: Definition. For any set $x$, a function $f$ is called ${\omega} $-Jonsson iff $f$: $^{\omega}x$ $\rightarrow$ x and whenever $y$$\subseteq$$x$ and ...
0
votes
1answer
37 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
3
votes
2answers
35 views

Unique Linear Map- Linear Algebra

Let $E = {e_1, . . . , e_n}$ be a basis for $\mathbb{R}^n$ , and let $v_1, . . . , v_n$ be arbitrary vectors in $\mathbb{R}^m$. Prove that there is a unique linear map $T : \mathbb{R}^n \rightarrow ...
1
vote
1answer
43 views

Absolute Value Property of Field of Real Numbers

I don't think my thought process is correct. Also, does 'if and only if' indicate that I should automatically resort to proof by contradiction? Show that ${|b|} \le {a}$ if and only if $ {-a} \le b ...
0
votes
1answer
19 views

What am I doing wrong here? Showing $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$.

Show $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$ where $a$ is invertible. What I did is: If $\text{Ord}_{N}(a)|k$ it is obvious. Suppose $a^k\equiv 1 \pmod N$. Not let us assume by contradiction ...
3
votes
1answer
28 views

Recognizing genuine proof obstructions

This is a meta question about mathematics. It is not inspired by an actual problem. Also, I'm not sure to what extent the distinction I'm drawing makes sense. Question: How can I decide if an ...
1
vote
1answer
29 views

Completeness Axiom Proof

Let $A_1$, $A_2$, $A_3$... be a collection of nonempty sets, each of which is bounded above. (a.) Find a formula for sup($A_1$$\bigcup$$A_2$). Extend this to sup($\bigcup^{n}_{k=1}$$A_k$). (b.) ...
1
vote
3answers
31 views

Prove that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$

Could someone please show me the proof that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$ I have no idea where to begin with this one. Thanks.
0
votes
3answers
34 views

Show that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$

Could someone please show me the algebraic steps in showing that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$? As the way I see it $\frac{1}{2}x^{\frac{2}{x}} ...
2
votes
0answers
53 views

A false conjecture by de Polignac

(This question would be similar to my other on Goldbach's conjecture so I'll change the "rules") In 1848 de Polignac claimed that "every odd integer is the sum of a prime $p$ and a power of $2$". For ...
1
vote
0answers
69 views
+50

Conjectured primality test for specific class of $N=k\cdot 6^n-1$

How to prove that this conjecture is true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ~\text{and}~ x ...
1
vote
0answers
23 views

On properties of linear orders

I have a simple question. Let A={a,b,c,...} be a set and > a total strict order on $2^A$. Total strict order means that for any two subsets of A, say S and S', either S>S' or S'>S but not both. The ...
1
vote
2answers
27 views

Rigorous Linear Transformation Proof

$T:V \rightarrow V$ We could also write: $T:V \rightarrow Im(T)$ The question tells us that $Im(T)=Im(T^2)$ It's intuitively obvious that this means that T then maps $Im(T)$ to itself so if you ...
3
votes
2answers
122 views

Is $f(x) = x^3 \sin \frac{1}{x} $ uniformly continuous on $(0, \infty)$?

Since the derivitive of $f$ is bounded on a neighborhood of $0$, $f$ is uniformly continuous on $(0, M)$ where $M$ is any positive number. I'd like to prove that $f$ is uniformly continuous on a ...
0
votes
1answer
20 views

If $\text{gcd}(a,561)=1$, then $a^{560}=1\mod 561$ [duplicate]

We have the factorization $561=3*11*17$. Because $\text{gcd}(a,561)=1$, there are integers $x,y$ such that $ax+561y=1$. So $ax=1\mod 561$. Since the gcd is $1$, we have $a\not \in 3\mathbb{Z}, ...
1
vote
0answers
32 views

Basic Set-Theoretic Properties from Halmos

I've been backtracking lately to make sure that I have a solid set-theoretic background before taking measure theory this fall. Here's a few facts I've come across today, and my attempted proofs. Let ...
0
votes
1answer
38 views

Proof for the coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$

Theorem: The coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$ is $\binom{2n-1}{n-1}$. How to prove this? Multinomial theorem produces the following $$ \left(\sum_{k=0}^{n} x^k \right)^n = ...
23
votes
2answers
819 views

A false conjecture by Goldbach

In 1752 Goldbach send this conjecture to Euler: "Every odd integer can be written in the form $p+2a^2$ where $p$ is a prime or $1$ and $a$ is a natural number (can be even 0)." This conjecture turned ...
2
votes
1answer
64 views

Can I assume that the dimension of a vector space is always non-negative?

I'm trying to prove that if $V$ is finite-dimensional and $U_1,...,U_m$ are subspaces of $V$, then $\dim(U_1+...+U_m)\le \dim U_1+...+\dim U_m$ through induction. For $m=1$, the inequality is trivial ...
1
vote
1answer
22 views

Zero of holomorphic function

Let $\Omega \subset \mathbb{C}$ be an open set that contains the unit ball $D$ and let $f \in \mathcal{O}(\Omega)$ a non constant map s.t. $|f(z)| = 1$ for all $z \in \partial D$. Show that $f$ has a ...
3
votes
1answer
73 views

Irrationality of ${5^{1/7}}$

I am struggling with elementary proofs, and would appreciate any feedback as to the logic and structure of my work. Show that ${5^{1/7}}$ does not represent a rational number. Suppose ${5^{1/7}}$ is ...
1
vote
2answers
78 views

Prove the Identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $

By considering the fact that $f(\pi/2)=1$, prove the identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $ This question was is a subsection in a chapter on Fourier series, can I use my ...
1
vote
0answers
35 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
1
vote
1answer
38 views

Proof strategy/writing for change of variables

Claim: If $f(x)=g(x)$ for all $x$, then $f(x+c)=g(x+c)$ for all x. Proof (attempt): Set $u=x-c$, and substitute $x=u+c$. $f(x)=g(x)$ implies $f(u+c)=g(u+c)$ for all $u$. Because $u$ is a dummy ...
2
votes
1answer
45 views

Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.

This is Velleman's exercise 3.4.26 (b): Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$. I do understand that a number will be ...
4
votes
2answers
58 views

Finding all solutions of $x^2+2x-15\equiv0 \pmod{105}$- Proof strategy.

Find all solutions of $x^2+2x-15\equiv0 \pmod{105}$. Now, I wanted to suggest a proof relying on the algorithm presented in class, and there are some parts where I could use some help or criticism. ...
-2
votes
1answer
51 views

Proof: A matrix with $m$ rows and $n$ colums has $nm$ entries.

How to prove rigorously the following statement: A matrix (a collection of numbers $a_{ij}:1\leq i \leq m, 1\leq j \leq n)$ with $m$ rows and $n$ colums has $nm$ entries. By rigorously I mean ...
1
vote
1answer
38 views

Requirement of a formal proof

There exists a continuous function $f$ whose domain is $[2,5]$ and the range is $(3,4)$. We have to prove that there exists at least one point $p \in (2,5)$ such that $f(p)=p$. Now this is easy to ...
2
votes
6answers
127 views

How to prove $\lim_{n\rightarrow\infty}nx^n=0$ without L'Hôpital's rule, where $x \in [0,1)$??

How to prove $$\lim_{n\rightarrow\infty}nx^n=0$$ without L'Hôpital's rule? where $x \in [0,1)$ and $n=1,2,3,...$. I know one of way to prove this is to treat $n$ is real, and $n$ and ...
0
votes
2answers
83 views

Prove that $\lim _{x \to \infty} \sin x$ doesn't exist (using delta epsilon)

though there is a question already asked in this site similar to this i want to prove that $\lim _{x \to \infty} \sin x$ doesn't exist using epsilon and delta. I don't know how to do this because ...
3
votes
2answers
123 views

Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.

Basically I need help in proving that if $U\supseteq \mathbb Q $ is an open set in $\mathbb R$ with the usual topology then $\mathbb R \setminus U$ is countable. I'm not really sure how to proceed. ...
1
vote
1answer
22 views

Determining Injectivity, surjectivity, bijectivity, and inverses

I was given a question that begins like this. Suppose that $A$ is the set $\{a,b,c\}$ (these are just names for some three elements - you don't know anything about $a,b,$ or $c$). Consider the ...
1
vote
2answers
54 views

Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$.

Here is the proof that I am currently working on. Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$. Hint: Apply the triangle inequality twice. Do not consider eight cases. I ...
1
vote
4answers
95 views

IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
4
votes
0answers
59 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
2
votes
0answers
18 views

The domain of the sum rule(probability: The logic of science)

Anyone read Probability Theory: The logic of Science. Please help, I've been really stuck for ages at how the sum rule has it domain derived and I don't have any teacher to ask. Question How is ...
1
vote
2answers
37 views

Is this function a bijection?

$f: \Bbb N \to P(\Bbb N)$ be given by $f(n) = \{n+1,n+2,n+3,\ldots\}.$ From general intuition and reasoning I think the function is not injective here is my work. If $n = 1$ $f(1) = ...
0
votes
2answers
38 views

Proof: $ A - (B - C) \subseteq (A - B) - C$

Question: Prove or disprove the following statements: For all sets $A, B, C$: a) $A - (B - C) \subseteq (A - B) - C$ b) $(A - B) - C \subseteq A - (B - C)$ c) If $A - (B - C) \subseteq ...
0
votes
2answers
43 views

Let $(s_n)$ be a convergent sequence of real numbers such that $s_n \neq 0$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty}s_n=s\neq 0$.

Prove that $\sup \{\frac{1}{|s_n|} : n \in \mathbb{N}\}>0$ Any help on getting this proof started would be appreciated. I know it must be related to proving that $\inf \{|s_n|:n \in ...
1
vote
5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
1
vote
0answers
32 views

The Polar-Coordinate Form of Cauchy-Riemann

Write $f(z) = u(r, \theta) + iv(r, \theta)$; suppose that the first-order partials of $u, v$ with respect to $r, \theta$ are continuously differentiable in some neighborhood of $z$ and satisfy ...