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
53 views

Doubt : Invariance in Geometry

I was working my way through some Proof Problems in Discrete Maths by Rosen, when I came across the following question: What Geometric proposition ( having an invariant property ) does this ...
2
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2answers
113 views

If $n^2$ is even $n$ is even

I understand that there are already several answers to how to prove this question: Prove if $n^2$ is even, then $n$ is even. Prove that if $n^2$ is even then $n$ is even I am trying to understand ...
2
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1answer
49 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
2
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2answers
52 views

Discrete Math Proof: $A \cup B$

I'm preparing ahead for a Discrete Math course coming up this year by doing some practice problems supplemented by online notes. The problem I'm having trouble proving is the following: $A \cup B ...
0
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2answers
52 views

What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...
4
votes
2answers
67 views

How to prove $( \sum_{n=1}^{\infty} |x_n|^2)^{1/2} \le \sum_{n=1}^{\infty} |x_n|$ (cauchy-product)

I am having this: $ (\ x_n)\ _{n \in \mathbb N} $ is sequence in $\mathbb C$, so the series $\sum_{n=1}^{\infty} |x_n|$ converges. I've already proved that the series $\sum_{n=1}^{\infty} |x_n|^2$ ...
0
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1answer
47 views

Next step to take in direct proof?

This is a problem from Discrete Mathematics and its Applications. I understand the basic ideas of the direct proof. Basically a proof is a conclusion from a series of steps to establish the truth of ...
2
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1answer
47 views

Next step to take in direct proof or a workaround around current dilemma?

This is a problem from Discrete Math and Its Applications I used a direct proof to do this proof. I understand the process/idea behind the direct proof, mainly (from ...
2
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1answer
63 views

$\exists x (P(x) \to \forall y P(y))$ [duplicate]

Prove $\exists x (P(x) \to \forall y P(y))$. Let x = y. Suppose P(x) is true. Let y be arbitrary. Since P(x) is true, it must be that P(y) is true. Since y was arbitrary, we can conclude that ...
3
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1answer
84 views

How to prove that $2^x,3^x,5^x\in\mathbb N$ implies $x\in\mathbb N$? [duplicate]

Let $x\in\mathbb R$ and suppose that $2^x,3^x$ and $5^x$ are all integers. Does it imply that $x$ is also necessarily an integer? I read somewhere that the answer is "Yes" and a proof is known, but I ...
0
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1answer
27 views

Cases for x in $ \forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x) $.

This is from Velleman p145, problem 28. Theorem: $\forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x)$. Author's Proof: Let x be an arbitrary real number. Case 1. $x = 0$. Let $y ...
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0answers
28 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
1
vote
2answers
63 views

Evaluating $\sum_{k=0}^n \frac{1}{(2k+1)!(2(n-k))!}$

Evidently: $$\sum_{k=0}^n \frac{1}{(2k+1)!(2(n-k))!} = \frac{4^n}{(2n+1)!}$$ (says wolfram alpha) But what is a good way to come up with this?
2
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7answers
125 views

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$ I know this is true because any even number that is squared will be even, is it also true than any even number ...
3
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1answer
42 views

How to derive this simple geometric relationship using cosine law?

Given the above figure, I need to show that $$cos(a_2) = \frac{x_1^2 + y_1^2 - L_1^2 - L_2^2}{2L_1L_2} $$ Where $L_1, L_2$ are the length of the red lines respectively, and $a_1, a_2$ are the ...
2
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2answers
92 views

Must proofs always be cited (Thesis)?

I have some proofs of theorems in my thesis that are very similar to the proofs from the literature ( "my" proofs are more extended and have more explaination, the structure isn't the same either). ...
1
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2answers
95 views

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn't be formal, but in the "usual" ...
3
votes
1answer
82 views

Convergence in dual of Sobolev space

Hi please view the following question: Consider Sobolev space $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is bounded. We also have a mapping $a: \Omega \times \mathbb{R} \times ...
0
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2answers
24 views

The minimal polynomial divides the characteristic polynomial

How do I show that the minimal polynomial divides the characteristic polynomial? I believe I need to use the Cayley-Hamilton theorem which I understand to be The characteristic polynomial of a linear ...
1
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1answer
72 views

Prove that a function is continuous at $x = x_{0}$ using the $\delta - \epsilon$ definition

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$ $\forall \epsilon > 0$, $\exists \delta = ?$ ...
0
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1answer
63 views

Prove that function $f$ is continuous at $x = x_{0}$

In class we're given the following definition about continuity, and I want to apply this definition to the problems that follow: $f$ is continuous at $x_{0} \in \mathrm{dom}(f)$ if $\forall x_{n} \in ...
2
votes
1answer
73 views

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$

Prove that $$f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$$ is continuous at $0$ and discontinuous everywhere else Proof: ...
1
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2answers
65 views

prove $f$ is a constant [duplicate]

Lets's say we have a differentiable function $f:[a,b]\to \mathbb{R}$ with $f^\prime\equiv0$ How do I show that $f\equiv C$ by using the mean value theorem?
3
votes
2answers
138 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
1
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2answers
54 views

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous. I know that the Dirichlet function is discontinuous everywhere ...
-2
votes
1answer
55 views

Can I tell whether this expression is positive?

Can I say that the following is greater than zero $$ \frac{2 \sqrt{xy} - y}{2(z+1)},$$ when $x \leq 2y \leq z$? What if, instead, $2y \leq z \leq x $? For the second condition i.e, $2y \leq z \leq ...
20
votes
5answers
1k views

Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
0
votes
1answer
24 views

An upper bound to this fraction

The following is an expression I am trying to upper bound by a constant $$I=\frac{x}{1+2y}\leq \ ?$$ The condition that I am using is $$ 2 x < y $$ I have tried the following $$ I = ...
1
vote
1answer
39 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
5
votes
5answers
191 views

Showing uniqueness of integers in base 3

I have recently begun self-studying Number Theory, and am working on proving: Show that every integer $n>0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and ...
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0answers
44 views

Showing an equality with sequences and sets

Let $a,b$ be two real numbers and $(a_n) \subset \mathbb{Q}$ be a decreasing sequence of rationals such that $a_n \to a $. Also, take a strictly increasing sequence $(b_n) \subset \mathbb{Q} $ such ...
2
votes
2answers
64 views

How to identify an error in a proof?

Right now I'm studying how to find errors in proofs by looking for common mistakes such as circular reasoning, using examples etc. I haven't had too many problems for the most part but I've run into a ...
3
votes
1answer
67 views

Density of a set around $0$ and on $\mathbb{R}$

In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$. Why is that enough to prove that it is dense on $\mathbb{R}$ ?
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2answers
133 views

how to show that a function is unbounded?

How to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded. I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$
1
vote
2answers
69 views

Proof strategy for basic proofs.

I'm currently in a discrete mathematics course and I'm having quite a bit of trouble with the idea of proofs. From what I understand the one I've been stuck on is also rather simple but to me it's ...
2
votes
3answers
101 views

Showing that a composite number has a small prime divisor?

At the moment I'm working on proving some statements and I've run into one that I can't seem to wrap my head around. It goes like this: For $n \in \mathbb{Z}^+$, we define $\sqrt{n}$ as the real ...
6
votes
2answers
171 views

Can you formalize the proof that $(1 + 2 + \cdots n)^2 = 1^3 + 2^3 + \cdots + n^3$ given here?

This website gives the following proof without words for the identity $(1 + 2 + \cdots n)^2 = 1^3 + 2^3 + \cdots + n^3$. I find it interesting but have trouble seeing the proof behind it. Could ...
2
votes
4answers
88 views

Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$

Show that $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$ Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$. ...
3
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4answers
75 views

Proof of divisibility using modular arithmetic: $5\mid 6^n - 5n + 4$

Prove that: $$6^n - 5n + 4 \space \text{is divisible by 5 for} \space n\ge1$$ Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using ...
5
votes
6answers
136 views

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$

Prove: If $a^2+b^2=1$ and $c^2+d^2=1$, then $ac+bd\le1$ I seem to struggle with this simple proof. All I managed to find is that ac+bd=-4 (which might not even be correct).
0
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2answers
54 views

Inference rule for Non-Empty Domains

I am currently experimenting with logic frameworks. I am basically using something along dependent types as in "Proof-assistants using Dependent Type Systems" by Henk Barendregt and Herman Geuvers. ...
17
votes
1answer
220 views

How are long proofs “planned”?

I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't ...
1
vote
1answer
100 views

Proof on integrating factors that are a function of one variable

Here's another in my long series of study questions on ODEs (and more to come). I want to prove the following: The (non-exact) equation: $$P(x,y)\,dx + Q(x,y)\,dy = 0$$ Admits an integrating ...
3
votes
3answers
96 views

Pythagorean type diophantine equation.

How to find all solutions to $$ a^2+b^2+c^2+d^2=e^2+2$$ where all variables $a$ to $e$ are positive integers and $e^2 \equiv 1 \mod 8$ I tried using parameterization similar to ...
0
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0answers
35 views

How can I prove these generalizations of Weierstrass?

I have thought of the following generalizations of Weierstrass' theorem, but I'm not sure about how to prove them. (1) Let $f:[a, +\infty[ \to \mathbb{R}$ continuous. Then, ...
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0answers
30 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
2
votes
3answers
84 views

Proving $ C = D $ from $ A \triangle C = A \triangle D$ [duplicate]

I have been working on one part of the proof where my aim is to show that $C = D$. I have been able to prove that $ A \triangle C = A \triangle D$. Is it reasonable to conclude $ C = D $ from $ A ...
3
votes
3answers
61 views

$f: X \mapsto X$ is a function with $f^n = id_X$ for a $n \geq 1$. Prove that $f$ is a bijection.

The problem above is easy to see with $n = 1$, because then every element of $X$ maps to itself and the function $f$ is obviously bijective. By $n = 2$ we have for every $x \in X$ one $y \in X | f(x) ...
1
vote
2answers
70 views

Proving sum of digits of $111111…^2$ is square of sum of digits of $11111…$

How do you prove that the sum of the digits of the square of a number comprised solely of ones is the square of the sum of the digits of that number? For instance, the sum of digits of $111^2$ is 9, ...
2
votes
0answers
79 views

Prove that: if $T$ is an irreducible linear operator then $T$ is cyclic

Let $T:V\to V$ be a linear operator on a finite dimensional vector space $V$. I need to prove that: If $T$ is irreducible then $T$ is cyclic My definitions are: $T$ is an irreducible linear ...