For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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0
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2answers
41 views

Check my argument that this sequence does not converge.

We want to show that $\{n^2+1\}$ does not converge. It's pretty clear that it doesn't converge, and this is only part of a true/false question so I don't really have to explain it, but I would like to ...
4
votes
4answers
366 views

Mathematical Induction Question, Proof Help [duplicate]

Prove using Mathematical Induction that for all natural numbers ($n>0$): $$ \frac 1 {\sqrt{1}} + \frac 1 {\sqrt{2}} + \cdots + \frac 1 {\sqrt{n}} \ge \sqrt{n}. $$ ...
-1
votes
1answer
32 views

Number of ways to order a set of interdependent tasks

I am aware that this question has been asked and answered before here. (Combinatorics/Task Dependency) I'd like some help understanding a part of the answer. Consider the graph shown there: ...
1
vote
1answer
73 views

prove if $b \geq a$, then $a^{b} \geq b^a$

I found that if b = a - 1, then $a^{b} \leq b^{a}$ and if a = b, then $a^{b} = b^{a}$ for obvious reasons. Now, i'm having a hard time figuring out how to prove that if $b \geq a$, then $a^{b} \geq ...
0
votes
3answers
61 views

prove $m^{m-1} < (m-1)^m$ for m > 3

I found that if m > 3 then $m^{m-1} < (m-1)^m$ for m > 3 seems to hold true for a lot of cases. Can someone prove this inductively ?
2
votes
3answers
69 views

Given the sequence $a_0=1, a_1=2, a_2=3, a_n=a_{n-1}+a_{n-2}+a_{n-3}$, prove by strong induction that for $n\geq 0, a_n \leq 2^n$

I've been trying to work this out for some time and I keep getting stuck. Here is what I have thus far: Base Case: $n=0 ; 1 \leq 1$ $n=1 ; 2 \leq 2$ $n=2 ; 3 \leq 4$ Induction hypothesis: ...
1
vote
1answer
56 views

How to write a general proof to prove that for all $m$, $m^n \geq n^m$

After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$. In other words, this can be generalized as: For ...
0
votes
1answer
49 views

Big Omega Proof for $5^n = \Omega(6^n)$

I got the Big-O ($\mathcal{O}$) proof but I am having troubles with the Big Omega ($\Omega$) proof. I am trying to prove $6^n$ is not the tightest bound for $5^n$. Is this logic true: $7^n$, $8^n$ ...
0
votes
3answers
75 views

Proof by induction $n^2 \geq n+1 \ \forall n \geq2$

I have to prove by induction that $n^2 \geq n+1 \ \forall n \geq2$. I have done the following reasoning: the base case is easy to verify; supposing that $n^2 \geq n+1 $ is true, we prove $(n+1)^2 ...
1
vote
2answers
53 views

$\mathbb{C}$ Forms a Vector Space Over $\mathbb{R}^2$ Proof Question

In my Mathematical Techniques course we've been talking about vector spaces, bases, etc. There is one problem however that I cannot get my head around and that is to prove that $\mathbb{C}$ can be ...
1
vote
0answers
45 views

Correct process for proof in graph theory.

I'm working on what I'm sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the ...
0
votes
3answers
102 views

Show that 1 is the supremum of $S = \{ x \epsilon R: x^2 < x \}$

I'm still new to proof writing so I was wondering if I could have a little help organizing my thoughts on this, I attempted this proof in a slightly oblong way that is probably not the standard, but I ...
0
votes
1answer
35 views

rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until…

"rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until you have an equation with $\sqrt{5}$ on the left and a ratio of two expressions involving $\sqrt{5}$ on the right." Ok..All i need to know is if i'm ...
0
votes
1answer
44 views

If $a \in \mathbb{N}$, prove that gcd$(a, a+2)$ is $1$ if $a$ is odd and $2$ if $a$ is even.

Once again the problem is: If 'a' is an element of N, prove that gcd(a, a+2) is 1 if 'a' is an odd number, and 2 is 'a' is an even number. I really have no idea on how to prove this, and I'm brand ...
1
vote
1answer
78 views

Proof about a subset of a metric space

Prove that a subset $A$ of metric subspace $(P, p')$ of metric space $(M, p)$ is open in subspace $(P, p')$, regarded as a metric space in its own right, if and only if there exists an open set $U$ in ...
0
votes
1answer
45 views

Introduction chapter Exercise Q3 from “How to Prove It: A Structured Approach”

The following question is from the book "How to Prove It: A Structured Approach" Second Edition. Theorem 3 : There are infinitely many prime numbers. Euclid's proof Introduction Chapter : Exercise ...
0
votes
3answers
60 views

Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
6
votes
4answers
1k views

There exist no integers for which $x^2-4y=2$

I am working on a new exercise in my textbook: $$\text{Prove that: (P): }\;\nexists \;x,y \in \mathbb{Z}, x^2-4\cdot y = 2 $$ I am stuck and I would really like to see a correct proof so I can move ...
2
votes
2answers
75 views

Induction Proof without Explictly Using The Induction Hypothesis?

I have encountered several problems where one can prove the desired result without actually needing the induction hypothesis. More specifically, you basically just pick $n \in \mathbb{N}$ and run ...
0
votes
3answers
134 views

Proving that the square root of 5 is irrational

Prove that $\sqrt{5}$ is irrational. I begin with the identity $(\sqrt{5} + 2 )(\sqrt{5} - 2 ) = 1$. Then I am told to extract $\sqrt{5}$ from the first or second factor and consider it to be ...
0
votes
1answer
17 views

Proving a statement with two variables by complete induction

I was recently introduced to this topic and I'm trying to prove Tue following statement. For most of numbers, m^n => n^m So I derived this into something that could be proved by induction... The ...
3
votes
4answers
133 views

How to prove which of two numbers written as powers is bigger?

Prove which number is larger: a) $10^{100!}$ or $10^{10^{100}}$ b) $e^\pi$ or $\pi^e$ I know we all know how to plug these into the calculator and check, but how someone mathematically prove which ...
1
vote
1answer
85 views

How to write a formal proof of the statement: For all real numbers $x$, if $x \ge 1$ then $\frac{3|x-2|}{x} \le4$

For all real numbers $x$, if $x\ge1$ then $\frac{3|x-2|}{x} \le 4$ I understand that I must algebraically show how to build on $x\ge1$ to reach $\frac{3|x-2|}{x} \le4$, but cant for the life of me! I ...
2
votes
1answer
78 views

Asymptotic Function proof?

I am doing questions from past exams and I stumbled upon this one. I have no idea how to go about solving it.I never had any logarithmic functions in my previous bigOh proofs nor have I had to use ...
0
votes
4answers
30 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
15
votes
3answers
569 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since ...
-1
votes
2answers
65 views

How to prove that for all $x$ there exists some $y$ where $ (x^2 + y^2 \geq 0)$?

How can I prove that $\forall \,\,x\,\,\exists y\,\,,(x^2 + y^2 \geq 0)$?
-1
votes
1answer
133 views

Prove a=v*dv/dx

Using calculus, and assuming a particle moving along the x-axis is concerned, prove that $a=v*dv/dx$ ~~~~~~~~~~~~ this is what I did, but im not sure it's rigorous enough: $a=dv/dt$ $t=x/v$ ...
1
vote
2answers
30 views

Proof regarding the function $\cos(1/x)$

Prove that for every number $a>0$ there exists 2 numbers $x,y$ with $0<x,y<a$ for which $f(x)>0$ and $f(y)<0$ with $f = \cos(\dfrac{1}{x})$. How do I go about proving this?
3
votes
1answer
37 views

Help explain existence of a limit point of a sequence implies infinitely many $m$ where $d(x,x_m)<\epsilon$

I don't understand the phrase "...all but finitely many elements...". What does this mean exactly and how does the conclusion "Infinitely many elementsof the sequence $\{x_k\}$ must also be within ...
1
vote
2answers
55 views

Help with a sequence proof problem

I have the following theorem to prove, and the book makes a certain suggestion that I don't understand. Theorem Suppose that the sequence $\{a_{n}\}$ converges to $l$ and that the sequence ...
0
votes
1answer
77 views

Find all natural numbers n such that n^2 < 2^n

Using induction proof, find all the natural numbers $n$ such that $n^2 < 2^n$. I know that $n$ does not work for $2, 3$, and $4$ but it does work for $0$ and $1$ as well as any number greater than ...
1
vote
1answer
55 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
0
votes
2answers
203 views

A New, Possible Proof of the Infinitude of the Primes?

$$1=1$$ $$2=2$$ $$3=3$$ $4=2\cdot2$ At $4$, the first prime number, $2$, is there as a factor. So I say that at the square of $2$, $2$ comes into play as a prime factor. At this point, $2$ is the ...
0
votes
1answer
25 views

rank propositional formula - exercise

Let $r$ be the rank function of a propositional formula, show that $r(\phi)<r(\psi)$ if $\phi$ is a proper subformula of $\psi$. I don't know how to prove it.
2
votes
0answers
43 views

Prove the big theta

I need to find a $n_0$ and $k$ for Big Oh and an $n_0$ and $k$ for Big Omega, to find a big theta bound for: $5n^2 - 9n = \theta(n^2)$ Can anyone help me and show me how to find these for this ...
3
votes
4answers
604 views

How do I make this simple proof better (and more correct?)

Let $x$ and $y$ be real numbers. If $x\cdot{y}>\frac{1}{2}$, then $x^2+y^2>1$. Proof: We will prove with the direct method. Let $x$ and $y$ be real numbers. Since $$ x\cdot{y}>\frac{1}{2} $$ ...
0
votes
1answer
125 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
0
votes
1answer
19 views

How to use the triangle inequality to get $d(x_k,y)\ge d(x,y)-d(x_k,x)$ (Prove limit of a sequence is unique)

Here, $d$ is the Euclidean distance: And the triangle inequality in terms of $d$ is: I have no idea how to come up with the $d(x_k,y)\ge d(x,y)-d(x_k,x)$ inequality. What I've tried: ...
1
vote
0answers
86 views

Proof of elliptic curves being an abelian group

What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements ...
1
vote
0answers
46 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
0
votes
1answer
78 views

Big Theta Proof Tightness

I found that $n_0 = 1 $ and $k=5$ for Big Oh, but I am somewhat confused on how to prove big omega as I have a negative sign in my expression. Furthermore by showing big oh and big omega, am I showing ...
0
votes
0answers
14 views

Does this part of an arithmetic progression have a name?

In this arithmetic progression - 11+30w, 11 is the initial term, 30 is the common difference, and w is what? I use the letter w because it is the first letter of the word whole, and I use w to ...
0
votes
1answer
22 views

Finite Arithmetic Progressions - Beginning and End Points

First, I want to express the integers 27,29,31,33, and 35 in the form of a finite arithmetic progression. Second, I want to express the integers 37,39,41,43,45, and 47 in the form of a finite ...
0
votes
6answers
77 views

What is the proof of $n^2 = 1 + 3 + 5 … (2*n - 1)$ [duplicate]

What is the proof of $n^2 = 1 + 3 + 5 + ... + (2\times n - 1)$? While I verified that this is true for small numbers, I am looking for a mathematical proof for all Natural Numbers .
2
votes
2answers
31 views

Proving sets implication using the method of contradiction

Suppose S and T are sets. Consider the following implication: If $A∩B=∅$ and $A ∪B = B$, then $A = ∅$. Prove the given implication by contradiction. So I have started by coming up with the negation: ...
6
votes
1answer
97 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: ...
0
votes
2answers
67 views

a proof of contradiction

I am wondering whether the following is a valid proof?
2
votes
3answers
73 views

$a > b+1 \Rightarrow a>x>b$?

If I have $a,b \in \mathbb R$ such that $$a > b+1 $$ It is assured that $\exists\space x \in \mathbb Z: a>x>b$ Does this property have some special name? How can this be proved? This idea ...
1
vote
3answers
148 views

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number. How do I prove this, and also which proving method would I use? I'm confused between that and ...