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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2
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4answers
74 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
2
votes
2answers
72 views

Proving by induction that any two natural numbers are equal.

This is something I've been working on for a while now; although it seems trivial, I am confused. I can't seem to find the error. Originally I thought the problem was with the base case, then I ...
0
votes
2answers
59 views

Proving that there is an irrational number between any two unequal rational numbers.

I'm trying to prove that there is an irrational number between any two unequal rational numbers $a, b$. Here's a "proof" I have right now, but I'm not sure if it works. Let $a, b$ be two unequal ...
2
votes
3answers
65 views

How can I prove a limit is infinity?

How can I prove that the $\lim \limits_{x \to 1^+} \frac{x^2}{x-1}=\infty$ using the $\epsilon -\delta$ definition of a limit? I think I start with $\forall$M>0, want $\delta$>0. After that I'm not ...
2
votes
2answers
30 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
1
vote
4answers
39 views

Using proof by contraposition to show that if $n\in\mathbb Z$ and $3n+2$ is even, then $n$ is even

I have my answer below but there is one step that I am not understanding...and maybe my brain is just not trained to understand it. Prove that if $n$ is an integer and $3n+2$ is even, then $n$ is ...
0
votes
2answers
44 views

How to prove that the Nested Interval Theorem fails to hold in $\mathbb Q$?

Claim: The Nested Interval Theorem does not hold in $\mathbb Q$. I can prove this by using sequences $a_n$ and $b_n$ where $a_n < b_n$ and they both converge for an $x$ which is any irrational ...
1
vote
1answer
12 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
1
vote
1answer
44 views

Proof of Fundamental Lemma of Calculus of Variations

Let me preface this question by saying I'm actually a physicist, not a mathematician, so a lot of the language I see you guys using here is over my head, so if you can keep it simple, that would be ...
0
votes
2answers
41 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
0
votes
0answers
53 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
0
votes
4answers
98 views

Why can't I prove this statement by simple induction? Sum of $1/2^1 + \cdots+ n/2^n = x$

I have to prove the following: $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{2 + n}{2^n}. $$ I am trying to prove this by simple induction. First, I proved that $P(1)$ ...
1
vote
2answers
23 views

How to prove $\omega$ bound without using limit?

How to show $n^{3.4} - 2015n^{2} + 3$ $\in$ $\omega(n^{3})$ without using limit? According to the definition of $\omega$, $f(n)$ $\in$ $\omega(g(n))$ if and only if $\forall c > 0$, $\exists n_0$ ...
1
vote
1answer
49 views

Prove $\int $ $(1/x)$ dx = $ln|x| + c$.

Note the domain we are working with is $x$ is all real numbers except $0$. My solution: Separate the question into two cases. (1) Prove that the left-side and the right-side of the equation are ...
1
vote
2answers
45 views

Prove $\mid\frac{2a}{b} + \frac{2b}{a} \mid \ge 4, \forall a,b \in \mathbb{R} - \left\{0\right\}$

I started by squaring both sides and proving: $(\mid\frac{2a}{b} + \frac{2b}{a} \mid)^2 \ge 4^2, \forall a,b \in \mathbb{R} - \left\{0\right\}$ My work: Consider: $(\mid\frac{2a}{b} + \frac{2b}{a} ...
1
vote
1answer
40 views

How to proove the following general form of proof

Suppose I have a statement $p(m,n)$ where $m,n \in \mathbb{N}$ that I want to proove. Suppose I have proofs of the following: $p(1,n)$ true for all $n \in \mathbb{N}$. $p(m,1)$ true for all $m \in ...
2
votes
1answer
39 views

$4x^2+1$ factors only into $4y+1$ primes

How can one prove that numbers of the form $4x^2+1$ can only be divided by primes of the form $4y+1$ (e.g. there is no $x$ for which $7$ divides $4x^2+1$)? On a quick lookup, the statement is given ...
2
votes
1answer
33 views

Is $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ bijective and find the inverse function!

$F$ is the set of the sequences in $\mathbb C$ and $R$ is the set of the series in $\mathbb C$. $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ Now $\sum_{j \in \mathbb ...
0
votes
1answer
25 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
1
vote
0answers
29 views

Complex number - prove an inequality

Question: Given that:$$z^n\tan\theta_0 + z^{n-1}\tan\theta_1 + z^{n-2}\tan\theta_2 + ... + \tan\theta_n = 3$$ And that $\theta_i \in (0, \frac{\pi}{4})$, prove that: $$|z| > \frac{2}{3}$$ ...
2
votes
1answer
41 views

Complex numbers - minimum value proof

Question: For:$$|z - z_1|^2+|z - z_2|^2+|z - z_3|^2+\cdots+|z - z_n|^2 = S$$ Prove that the minimum value of $S$ is when:$$z = \frac{z_1+z_2+z_3+\cdots+z_n}{n}$$ I have no idea how to even ...
0
votes
1answer
37 views

Base 10 proof strategy

Let $a$ be a number written (in base 10) as $$a=a_0\cdot10^0+a_1\cdot10^1+a_2\cdot10^2+\cdots+a_n\cdot10^n$$ where $0\leq a_i <10$. Prove the following: 2 divides $a$ if and only if 2 ...
1
vote
2answers
61 views

How tot start proving $A \times B \times C \ne (A \times B) \times C$?

This is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the ...
4
votes
2answers
109 views

Second Grade Homework Problem - Methodology

Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem. Using only the natural numbers 1 through 9 ...
0
votes
1answer
52 views

Property of continuous functions regarding maximum

Claim 1: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ assumes a maximum value I know there's a theorem that states if $f$ is a continuous real-valued function on a closed interval ...
0
votes
1answer
58 views

Proof strategy for simple 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 ...
3
votes
1answer
34 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
4
votes
0answers
74 views

Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
0
votes
0answers
19 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
0
votes
4answers
33 views

Proof that composition of invertible linear transformations is invertible (without determinants)

A crucial concept in linear algebra is that the composition of two invertible linear transformations is itself invertible. Here is the first proof I learned of this fact: Proof: Suppose that $T_1: ...
2
votes
2answers
56 views

Closed form solution and combinatorial proof.

First of all, I would like to figure out a closed form solution for the following summation: $$\sum^{n}_{k=0} C(n,k)\cdot C(2n,n+k)$$ Where C(n,k) means n choose k, or $\frac{n!}{(n-k)!\cdot k!}$ ...
2
votes
1answer
28 views

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order. A partial order is a binary relation that is reflexive, anti-symmetric and transitive. So if $R$ is a partial order, ...
0
votes
1answer
52 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
votes
2answers
94 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
votes
1answer
47 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
votes
2answers
50 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
votes
2answers
38 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 ...
3
votes
2answers
64 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
votes
1answer
37 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
votes
1answer
44 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
votes
1answer
59 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 ...
0
votes
0answers
36 views

Cyclic convex quadrilateral property

Let $ABCD$ be a cyclic convex quadrilateral, and let $P$ be the intersection of the diagonals. Show that $\frac{PB}{PD}=\frac{AB}{AD}\cdot \frac{CB}{CD}$. I guess I need to use Ptolemy's ...
3
votes
1answer
77 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
votes
1answer
23 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 ...
1
vote
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
62 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?
1
vote
7answers
107 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
votes
1answer
37 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
votes
2answers
69 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
vote
2answers
87 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" ...