Tagged Questions

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
votes
1answer
15 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
5answers
104 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 ...
0
votes
1answer
59 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
65 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
27 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 ...
13
votes
3answers
315 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 ...
0
votes
2answers
58 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
49 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
25 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
27 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
39 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
39 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
49 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 ...
-1
votes
2answers
155 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
15 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
28 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
580 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
26 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
16 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
30 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
25 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
53 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
13 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
18 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
74 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
29 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
73 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
48 views

a proof of contradiction

I am wondering whether the following is a valid proof?
2
votes
3answers
72 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
35 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
34 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 ...
0
votes
1answer
50 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
38 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 ...
1
vote
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
39 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
70 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 ...
1
vote
1answer
144 views

For every $x\in\mathbb R$ and $\varepsilon$ > 0 , there exist $\,q,q'\in\mathbb Q$, such that $q<x<q'$ and $\left |q-q' \right |< \varepsilon$

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
vote
3answers
31 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) ...
2
votes
1answer
115 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
4answers
69 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
47 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
55 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
51 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 ...
-1
votes
1answer
75 views

Finding a minimum weight spanning tree? [duplicate]

Letting W be the weighted graph created by taking a complete graph K5 on five vertices 1, 2, 3, 4, 5 with the weight of each edge {x,y} given by ({x,y})=x+y, How would I find a minimum weight ...
2
votes
1answer
49 views

Verify my proof: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $

Could someone verify my proof and my proof-writing? Proposition: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $ Proof: Suppose $ y $ is any positive natural ...
0
votes
1answer
26 views

Proof with Cartesian coordinates.

Let $S_b := \{(x,y) \in\mathbb R^2 | y = 3x + b\}$ where $b\in\mathbb R$. Give a direct proof that if $(r,s)\in\mathbb R^2$, then there exists a $b\in\mathbb R$ such that $(r,s) \in S_b$. I have ...
3
votes
1answer
85 views

Uniform convergence on an interval

Let $a<c<b$. Let {$f_n$} be a sequence of functions converging uniformly on $[a,c]$ and $[c,b]$. Prove that {$f_n$} converges uniformly on $[a,b]$ My attempt: Intuitively, I see that {$f_n$} ...
2
votes
2answers
308 views

Prove that if a and b are positive integers, then there exists integers x and y such that 1/lcm(a,b)=x/a+y/b

My professor has not taught us the technique of writing proofs, he just continues to do them for us in class. So I am really stumped on this proof. Any help is greatly appreciated!
0
votes
0answers
17 views

Using a direct proof to prove circumscribed shapes.

I am looking at this problem: Use a Direct proof to show that if A is a circle circumscribed by a square B, and the square B is circumscribed by a Circle C, then the area of Circle C is twice the ...
0
votes
3answers
113 views

How would I show that Tn=3^n + 2 is a solution to the recurrence?

Would anyone be able to help me or give me some advice on the following problem: Consider the recurrence with $T_0 = 3$ and $T_{n+1} = 3T_n - 4$ for all $n \in \mathbb{N}$. How would I show that ...