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
49 views

Establish the convergence or divergence of a sequence [duplicate]

Establish the convergence or divergence of the sequence (y_n), where: y_n := 1/(n+1) + 1/(n+2) + ... = 1/(2n) for n /in N.
2
votes
2answers
70 views

Proving $a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$ without Euler's Theorem

Is there a different way of solving $$a^{(p-1)p^{k-1}} \equiv 1 \pmod {p^k}$$ without the use of Euler's Theorem (or proof of Euler's theorem for $p^k$)? I've tried to use the Chinese Remainder ...
0
votes
0answers
35 views

Corollary of the inverse function theorem

Let $U\subset \mathbb{R}^{n}$ and $ f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...
1
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0answers
22 views

Do nth degree polynomials derived using Least Squares Interpolation always have n+1 intersections with the function?

I have recently studied Interpolation Techniques in my College Numerical Methods class and I have this question: If we have a function $f(x)$ and we are asked to use Least Squares Interpolation(LSI) ...
0
votes
1answer
26 views

Product of a prime and other number expressed in two ways are equal

Let $p \in \mathbb P$ be an odd prime and let $1\leq a \leq p-1$ be such a number that $$a p = \left\lceil \sqrt{a p}\right\rceil ^2-\left\lceil \sqrt{\left\lceil \sqrt{a p}\right\rceil ^2-a ...
2
votes
4answers
186 views

Construction of a cauchy sequence

Construct a cauchy sequence $(r_n)$ such that $r_n\in\mathbb{Q}$ $ \forall n\in\mathbb{N}$ , $(r_n)$ is a Cauchy sequence but $\lim_{n \to \infty}r_n $ does not belong to $\mathbb{Q}$ Can anyone ...
1
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0answers
32 views

Fundamental questions about Logarithm and finding quadratic roots

Define: $(e^{iz}+e^{-iz})/2= cos z$ where $z \in \Bbb C $, i.e, the cosine function is defined for complex $z$. Now, is it true that for each $w \in \Bbb C $ there is $z \in \Bbb C $ such that $cos z ...
0
votes
1answer
37 views

Prove a consequence of the multivariable version of the inverse function theorem

The exercise is the following: Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ that is class $C^{1}$ such that there exists $c >0$ such that $$|f(x) - f(y)| \ge c|x-y|$$ for all $x,y \in ...
3
votes
5answers
85 views

Supppose $ f : X \rightarrow Y $ is a function and $ A \subseteq X $ is countable. Then $ f(A) $ is countable.

Supppose $ f : X \rightarrow Y $ is a function and $ A \subseteq X $ is countable. Then $ f(A) $ is countable. I am in an introduction to proofs course... we are studying topology in order to use ...
0
votes
1answer
25 views

Full proof of this matrix propierty

I have to prove the following question: ? I asked yesterday and got answers like "Since $A=Y^{-1}XY, A^2=(Y^{-1}XY)(Y^{-1}XY)=Y^{-1}X^2Y$. So $\alpha A^2+\beta A+γI=Y^{-1}(\alpha X^2+\beta ...
0
votes
2answers
28 views

Stuck on Circles question on tangents

Here the length of AO is equal to diameter of circle. AB and AC are tangents from A. The triangle ABC has to be proved equilateral. I put it in geogebra and it was indeed equilateral. I can't find ...
-1
votes
1answer
51 views

How to prove the inductive step in this Mathematical induction problem?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 6, pg 342]. Problem: a) Determine which amounts of postage can be formed using just $3$-cent and $10$-cent ...
2
votes
2answers
36 views

How to come up with relation in induction hypothesis for strong induction

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 2, page 341]. Problem: Let $P(n)$ be the statement that a postage of n cents can ...
2
votes
4answers
61 views

What is the trick to identify which of these are true

I was hoping to not have to create a bunch of fictional sets so that I can solve this problem. Any trick or rules to this? The sets A and B are subsets of a universal set U. Which of the following ...
0
votes
1answer
126 views

How to show the inductive step of the strong induction?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 341]. Problem: Use strong induction to show that all dominoes fall in an infinite arrangement of dominoes if ...
0
votes
2answers
46 views

How to get $k^{k + 1} + k^k$ to equate $(k+1)^{k+1}$?

This is a problem from Discrete Mathematics and its Applications Let $P(n)$ be the statement that $n!<n^n$, where $n$ is an integer greater than $1$. $\quad(a)$ What is the ...
1
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1answer
33 views

Show that $\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$

The Problem Show that $$\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$$ My Work $$\begin{align*}\frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k} &= ...
2
votes
3answers
39 views

How to show no other elements besides $\pm 1$ will be in the kernel of $h: \mathbb Z_p^* \rightarrow \mathbb Z_p^*$; $h(\bar{a}) = \bar{a}^2$.

Let $p$ be a prime and let $h: \mathbb Z_p^* \rightarrow \mathbb Z_p^*$ be defined by $h(\overline{a}) = \overline{a}^2$. Since $h(\overline{xy}) = \overline{xy}^2 = \overline{x}^2 \overline{y}^2 = ...
1
vote
2answers
32 views

How can this be proven (Matrices)

I need to prove why the image on the bottom is true, btw this is on a matrices unit so you know that the order of multiplication does matter
0
votes
1answer
43 views

How do I prove that $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ is a PID?

I'm curious how to prove $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ is a PID. Before I get started proving this, I want to know a correct direction. Is it a good way to prove this by showing that the ring ...
1
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1answer
24 views

Using the sequential Criterion, give a proof that $\lim\limits_{x\to 0} f(x)$ does not exist, where: $f(x) = -1$, $x \leq 0$ or $x$, $x>0$

Using the sequential Criterion, give a proof that $\lim\limits_{x \to 0} f(x)$ does not exist, where $$f(x) = \begin{cases}-1 & \text{if } x < 0\\ \ \ \ x & \text{if } x \geq 0 ...
0
votes
2answers
44 views

Reciprocal squares sum inequality [duplicate]

What is the easiest (preferably inductional) way without approximation of the sum_ to prove the following inequality: $\frac{1}{1^2}+\frac{1}{2^2} + \ldots +\frac{1}{n^2} \le 2 - \frac{1}{n}$
1
vote
1answer
40 views

Proving a function is convex

From the Defintion of convex: Theorem to be proven: If $f$ is differentiable and $f'$ is increasing, then $f$ is convex. Use Proof by Contradiction. Consider, $I = (a, b)$ with $a < x < ...
1
vote
1answer
48 views

Cayley transformation of a skew-symmetric matrix is orthogonal?

If $S$ is skew-symmetric ($S^{T} = -S$), how do I show that $Q$ is orthogonal where $$Q = (I + S)(I - S)^{-1}$$ which is the Cayley transformation of $S$.
0
votes
0answers
10 views

Estimation of changes in solution x when A change

Suppose I have system $Ax = b$ where A = [${2}$ $-1$ $1$; $-1$ $10^{-10}$ $10^{-10}$; $1$ $10^{-10}$ $10^{-10}$]; b = [$2(1 + 10^{-10})$; $-10^{-10}$; $-10^{-10}$] and x = [$10^{-10}$; $-1$; $1$] ...
2
votes
1answer
27 views

I cant identify the quantifier

For a simple question like Let x, y ∈ Z. If 3 | x or 3 | y then 3 | x y, Is it alright to assume all x and all y exist in Z? I am trying to negate the statement but since it does not say 'each' ...
0
votes
1answer
31 views

Negate the following statement

“The integer n is even if and only if$$ n^2 + 1$$ is even"" The professor wrote that the negation of this statement is "The integer n is even if and only if $$ n^2 + 1$$is odd." I am pretty sure ...
1
vote
1answer
30 views

Proof in which sup A is related to inf B

Let $A \ne \emptyset$ also $A \subset [1,3].$ Define $B$ to be the set of positive real numbers $x$ such that $\sqrt{x}-1$ is an upper bound of $A.$ Prove that $\inf B = (1 + \sup A)^2.$ Here are my ...
2
votes
1answer
17 views

Proof: Number theory: Prove that if $n$ is composite, then the least $b$ such that $n$ is not $b$-pseudoprime is prime.

I'm looking to prove this, but not too sure how: If $n$ is composite, then the least $b$ such that $n$ is not $b$-psp is prime. Thanks!
0
votes
1answer
21 views

Deduce the Nested Interval Property from MCT?

Let $a_n$ be an increasing sequence, $b_n$ be a decreasing sequence, and assume that $a_n < b_n, \space \forall n \in N$. Show that $\lim(a_n) ≤ \lim(b_n)$, and thereby deduce the Nested Intervals ...
0
votes
1answer
31 views

If $x_n$ and $y_n$ are sequences where $x$ converges to a value other than $0$ and $x_ny_n$ converges, then $y_n$ converges.

Does this make sense? Since $x_n$ converges to a value other than $0$, and $x_ny_n$ converges, then: $$y_n = \frac{x_ny_n}{x_n}$$ also converges.
1
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0answers
20 views

Proof: Let $\gcd(a,m)=1$. Then $a^i\equiv a^j\pmod{m}\iff i\equiv j\pmod{\text{ord}_m a}$

Would someone be so kind as to look over my proof for me? $\underline{\implies}$ EDITED Assume $a^i \equiv a^j\pmod m$. Then, $a^{i-j} \equiv 1 \pmod m$. This means that $i-j = k \cdot ...
0
votes
1answer
38 views

Contrapositive of this statement

Suppose $∀x ∈ R, ∃y ∈ R$, s.t. $∀z ∈ R.$ Consider the following statement: $$z > y \implies z > x + y $$ The contrapositive of this statement is: $$z ≤ x+y \implies z ≤ y$$ with the same ...
1
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3answers
47 views

Defining prime numbers for proofs

In my discrete mathematics book under existence proofs it has Prove that there exists a prime $p$ such that $2^p -1$ is composite. It then goes on to say by trial and error we find $2^{11}-1$ ...
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votes
1answer
46 views

Show that $\lim (\sqrt{n^2+1)}-n) = 0$ [duplicate]

Can't use limit rules as $\sqrt{n^2+1}$ and n are not convergent sequences
1
vote
1answer
24 views

Prove that if $\displaystyle \lim_{n\to \infty} x_n = x$ and if $x > 0$, there exists a natural number $M$ such that $x_n >0$ for all $n > M$.

Prove that if $\displaystyle \lim_{n\to \infty} x_n = x$ and if $ x > 0$, then there exists a natural number $M$ such that $x_n >0$ for all $n > M$. Is this not just a proof of the ...
0
votes
1answer
27 views

Definition of Ordinal (w/ Axiom of Regularity) (problem 37, page 208, Enderton's Elements of Set Theory)

Given the definition of an ordinal to be well-ordered by $\in$ and transitive, I am interested with proving the following: I know and understand the following which easily proves it, but uses the ...
7
votes
4answers
183 views

Showing $(P\to Q)\land (Q\to R)\equiv (P\to R)\land[(P\leftrightarrow Q)\lor (R\leftrightarrow Q)]$ {without truth table}

Problem: Show $(P\to Q)\land (Q\to R)\equiv (P\to R)\land[(P\leftrightarrow Q)\lor (R\leftrightarrow Q)]$ Source: As was noted in the original post, this problem is from Daniel J. Velleman's book ...
1
vote
1answer
21 views

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$ How can I do this? I'm kinda stuck with this 2-norm of the matrix. If it would be $\|A\|_{\text{frob}}$ then ...
3
votes
1answer
52 views

A question on proving the uniqueness of a mathematical object

When proving that there is a unique mathematical object that satisfies a particular condition, e.g., the inverse of an element of a group, is the intuition behind it the following? You assume that ...
1
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2answers
39 views

Prove identity related to nths root of unity

If $1=z_0,z_1,...,z_{n-1}$ are nth roots of unity, prove that $$(z-z_1)(z-z_2)...(z-z_{n-1})=1+z+z^2+...+z^{n-1}$$ I don't know what is meant by the condition given. If I substitute ...
1
vote
2answers
90 views

If $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $ f'(x) \leq f(x), \forall x,$ then $f\equiv 0 \ ?$

Suppose $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $ f'(x) \leq f(x), \forall x.$ Could anyone advise me how to prove/disprove that $f \equiv 0 \ ?$ Thank you.
1
vote
1answer
58 views

Prove that every minimal edge cut in a simple connected graph $G(V,E)$ is even, then $G(V,E)$ has no odd degree vertex.

I need to prove/disprove the following statement -- If every minimal edge cut in a simple connected graph $G(V,E)$ is even, then $G(V,E)$ has no odd degree vertex. I am a bit confused about one of ...
0
votes
1answer
29 views

why does $\varphi'(N)=0$ in this proof?

Fulton's book on page 105 defines $N$: Afterwards Fulton writes this solution for this lemma: I didn't understand why $\varphi'(N)=0$ Thanks
1
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1answer
42 views

Any advantage when proving linear algebra statements without using bases?

I always felt that proofs in linear algebra that do not assume the existence of bases seem more elegant but is there also something mathematically more valuable about these proofs? I know that the ...
0
votes
4answers
65 views

In the Collatz function, why does $2^k-1$ reach $3^k-1$ after $2k$ steps, and could it be used to find divergent trajectories?

If you start calculating the Collatz function from an integer of the form $2^k-1$, you will reach $3^k-1$ after $2k$ steps. So, it is possible to pick a starting value that continuously zig-zags ...
1
vote
1answer
93 views

A question regarding a small calculation in the proof of a theorem.

I understand the theorem overall, and that the $\epsilon$ values are arbitrary. Notice how the values $\epsilon_1=\frac\epsilon{2M}$ and $\epsilon_2=\frac\epsilon{2(|t| + 1)}$ are just assumed, and ...
1
vote
1answer
54 views

Two rows or two columns with the same number of plusses

I have tried drawn numerous tables in attempt to explain this and understand that the number of cells must be even however, I am not sure how to create this proof. I appreciate your support. Each ...
3
votes
1answer
28 views

Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent

Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent My professor's proof is as follows: So we know that the radius of convergence is $R = 1$. Now ...
0
votes
1answer
49 views

Show that $\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x \in\ X\}$

Let $X$ be a nonempty set, and let $f$ and $g$ be defined on $X$ and ave bounded ranges in $\mathbb{R}$. Show that: $$\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x ...