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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3answers
38 views

How to prove which number 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 ...
2
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1answer
30 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
1answer
26 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 ...
3
votes
4answers
165 views
+100

If all mappings $f: A\to B$ are many-to-one, does there exist surjective $g: A\to B$?

Suppose sets $A$ and $B$ [edit: for $B\ne \emptyset$] are such that all mappings $f: A\to B$ are many-to-one (i.e. not injective). Can we prove that there must exist a surjective $g: A\to B$? Ideally, ...
0
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4answers
19 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 ...
0
votes
3answers
70 views

Limit of $a_n = -\cos \left( \frac{\pi}{8n-2} \right)$ (proof)

Warning to anyone who stumbles upon this: It is wrong completely and utterly don't use it for reference, thank you Don and Gerry for helping me see this So my first question is asking whether or not ...
2
votes
2answers
67 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 ...
-2
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0answers
58 views

Where Are the Primes In Relation To the Perfect Squares? How Are the Perfect Squares Arranged Along the Natural Number Line? [on hold]

The question is concerning the location of any given prime which satisfies Legendre's conjecture, or simply, any given prime. Do they not all? All primes > 3 are in the pair of arithmetic progressions ...
0
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2answers
47 views

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

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

How to prove a function is not onto?

Let $f : Z\to Z$ be the function defined by $f(x) = 3x + 1$. Prove that $f $ is not onto, using a proof by contradiction. (Choose an integer $n$, and then prove ($\forall m \in Z$)($f(m) ≠ n$) by ...
-1
votes
1answer
39 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: $a=dv/dt$ $t=x/v$ ...
5
votes
1answer
52 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: ...
7
votes
2answers
1k views

Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$

I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
1
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2answers
23 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?
1
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1answer
43 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
137 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 ...
1
vote
3answers
119 views

Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
3
votes
1answer
22 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
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2answers
31 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 ...
3
votes
4answers
575 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
33 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 ...
0
votes
1answer
14 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
26 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 ...
0
votes
1answer
41 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
1answer
17 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
14 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
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0answers
23 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
21 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
0answers
12 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
15 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
70 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
27 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: ...
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votes
2answers
41 views

a proof of contradiction

I am wondering whether the following is a valid proof?
2
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3answers
69 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 ...
0
votes
1answer
23 views

Show that $f'_+(a)=f'(a+)$ if both quantities exist.

Show that $f'_+(a)=f'(a+)$ if both quantities exist. I'm not really sure where to start, any help is appreciated. I came up with this: If $f'(a^+)$ exists, then by definition $f'(a+) = \lim_{x\to ...
1
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3answers
30 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 ...
0
votes
1answer
28 views

Show that | and $\downarrow$ are the only binary connectives \$ such that {$} is functionally complete.

I've been reading and coping with van Dalen's Logic and Structure for a a few days. However, I've getting problems to solve his Exercise 6 from Ch 1 Sec 1.3 (p.28). In this exercise, van Daken asks ...
0
votes
1answer
33 views

Proof on existence of the natural numbers, crucial step.

I am trying to understand/reconstruct the proof given by my Professor addressing the existence of natural numbers. However there is one step in particular I don't understand and the more I think about ...
2
votes
2answers
139 views

Critique this proof on compactness.

Problem: Prove or disprove, the metric space $X$ containing infinitely many points with the discrete metric is compact. Write a proof in the language of sequences and covers Proof: Take $(1/n) \to ...
1
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1answer
121 views

Proof that for every $\epsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |< \epsilon$

I'm asked to prove that for every $\varepsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |<\varepsilon$ where $x$ is a real number. ...
1
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2answers
24 views

How to show that the limit of $x(1+\sin(x)$ is not $\infty$ as $x\rightarrow \infty$?

Let $f(x)=x(1+\sin (x))$. The definition of $f$ tends to infinity as $x$ tends to infinity is: For any $M>0$, there exists an $X$ such that, for all $x\ge X$, $f(x)\ge M$ So, the negation of ...
0
votes
2answers
37 views

Proving that $x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$

How would you prove that if $x$ is an integer, then $$x-\left\lfloor \frac{x}{2} \right\rfloor\; =\; \left\lfloor \frac{x+1}{2} \right\rfloor$$ I tried to start by saying that if $x$ is an even ...
5
votes
3answers
68 views

Grade this proof of a surjective map from $\mathbb{R}^3$ to $\mathbb{R}^3$.

I just received this homework proof back in my abstract algebra class with a grade of 20%. I feel very cheated, to say the least. I present it here verbatim for your critiques. Please tell me what ...
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0answers
32 views

$a$, $b$, $c$ > 0, if $\frac ab + \frac bc + \frac ca$ is an integer, prove that $abc$ is a cube of an integer [duplicate]

$a$, $b$, $c$ are not equal to $0$, if $\frac ab$ + $\frac bc$ + $\frac ca$ is an integer, prove that $abc$ is a cube of an integer. I really have no idea, appreciate any kind of help
0
votes
1answer
17 views

Cartesian product proof with counterexample

I was asked to disprove the following statement by counterexample: Let A, B and C be sets. If A x C = B x C then A = B I was under the impression that: (x1, y1) = (x2, y2) if and only if x1 = x2 and ...
1
vote
3answers
27 views

Show that for all real numbers a and b, ab <= (1/2)(a^2+b^2)

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b$ $\epsilon$ $R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
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votes
4answers
60 views

There exists a real number $x$ such that if $x^2 ≥0$ then $ x=0$.

I have to prove: There exists a real number $x$ such that if $x^2 ≥0$ then $x=0$. I have no idea what should I proceed. I tried to come up with the contrapositive, and it doesn't help. I have this ...
1
vote
2answers
35 views

Using logical Properties to prove a tautology

So I have to prove this as a tautology. I've been stuck on this forever and am not sure where to go. I experimented and got this far, and looking for some pointers on where to take it next. (p → q) ...
0
votes
1answer
36 views

Practice Examples of Proofs by Induction, Direct/Indirect Method

I'm learning about proofs in school, quite a few different sorts (but not geometry ones), but the teacher is teaching by slides mainly, not books. The main ones are proof by ...
4
votes
1answer
50 views

Help getting started on a proof

I honestly have no idea where to start with the following proof, and I was wondering if anyone could help me get started. I don't want the whole idea, I just need to know where to start with this ...