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.

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1answer
70 views

Is this a valid proof of this math challenge problem?

From a fixed point P not in a given plane, three mutually perpendicular line segments are drawn terminating in the plane. Let a, b, c denote the lengths of the three segments. Show that ...
0
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2answers
45 views

Weird question about natural numbers. Obvious or not?

Given any subset $A,C \subset \Bbb{N}$, there exists a maximal subset $B \subset \Bbb{N}$ such that for all $b \in B, a \in A, \ |b - a| \in C$. For instance $A = \{3,5\}$, $C = \{2,4\}$, then ...
19
votes
4answers
1k views

We all use mathematical induction to prove results, but is there a proof of mathematical induction itself?

I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like $$1+2+3+\cdots+n = ...
0
votes
1answer
25 views

Show that (c,a)=(c,b)

In my book I have the implication: If $gcd(a,b)=1$ and $c|(a+b)$, then $gcd(c,a)=gcd(c,b)=1$. It gives me a hint that begins by supposing that $gcd(a,c)=gcd(b,c)=d$. But in my opinion, I do not ...
1
vote
1answer
31 views

Find all primes $p$ for which $x^2+2x+4\equiv 0 \pmod p$ is solvable. Am I correct?

Getting ready for an exam, I would like to focus on the correctness of my solution, final results and assumptions, and would appreciate any comment regarding it or even additional ...
3
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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
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1answer
9 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
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2answers
55 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
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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
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1answer
37 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
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1answer
35 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
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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
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4answers
114 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
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0answers
31 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
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0answers
54 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
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1answer
40 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
37 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
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1answer
44 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
20 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
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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
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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
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3answers
32 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
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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}} ...
0
votes
1answer
52 views
+50

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
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 ...
2
votes
0answers
123 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
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0answers
25 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
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2answers
28 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
124 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
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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
39 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
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2answers
840 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
65 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
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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
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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
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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
59 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
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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
55 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
98 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
60 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 ...