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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191 views

Do you need to find the domain of a function to prove injectivity?

I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" ...
4
votes
2answers
109 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
4
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3answers
75 views

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction: How to prove one of them?

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction How to prove one of them ? On Proofwiki there is an article proving the equivalence of the ...
4
votes
4answers
141 views

Prove that $A \mathop \triangle C \subseteq (A \mathop \triangle B)\cup (B \mathop \triangle C)$

Suppose A, B, and C are sets, prove that $A \mathop \triangle C \subseteq (A \mathop \triangle B)\cup (B \mathop \triangle C)$ I'm just wondering if this proof is ok, or if I'm overlooking something, ...
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2answers
6k views

Prove: If a sequence converges, then every subsequence converges to the same limit.

I need some help understanding this proof: Prove: If a sequence converges, then every subsequence converges to the same limit. Proof: Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k ...
4
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1answer
12k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
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4answers
64 views

Prove that $a \equiv b \pmod{m_1m_2}\implies a \equiv b \pmod {m_1}$

So, I have this problem: if $$a \equiv b \mod(m_1m_2)$$ then (show) $$a \equiv b \mod(m_1)$$. I have to do a proof, but I have no idea where to begin the proof. Can someone help? Proof (Edit): ...
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4answers
99 views

What is the definition of a labeled function?

I always see that people label their functions by giving an index. Specifically I have this example: $Theorem$: There is a unique binary operation $+:\mathbb{N}\times\mathbb{N}$ that satisfies the ...
4
votes
2answers
158 views

Why does this step work in this proof?

I'm trying to learn discrete math and am brushing up on proofs by reading Richard Hammack's Book of Proof. I'm tripped up on this proof... I understand that it's contrapositive, and why contrapositive ...
4
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3answers
303 views

If a graph of $2n$ vertices contains a Hamiltonian cycle, then can we reach every other vertex in $n$ steps?

Problem: Given a graph $G,$ with $2n$ vertices and at least one triangle. Is it possible to show that you can reach every other vertex in $n$ steps if $G$ contains a Hamilton cycle (HC)? EDIT: ...
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3answers
334 views

Set Theory: Polynomial Relations

I'm having a bit of trouble understanding exactly what this question is asking me in my Sets and Proofs homework: If a polynomial p over $\mathbb{R}$ is an expression of the form $p(x) = a_nx^n + ...
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votes
5answers
769 views

Mathematical induction equation involving a sum of binomial coefficients

I have a problem with a mathematical equation. I don’t find the given solution. This is the equation: $\sum\limits_{k=2}^{n-1} {k \choose 2} = {n \choose 3} $ I should show with induction that the ...
4
votes
1answer
155 views

Is this proof wrong?

For $n>1$, let $a_1, a_2, \dots, a_n$ be $n$ distinct integers. Prove that the polynomial $$f(x)=(x-a_1)(x-a_2)...(x-a_n) - 1$$ cannot be written as the product of two nonconstant ...
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4answers
193 views

Proving an inequality

Suppose $a$ and $b$ are real numbers. Prove that if $a<b$ then $\frac{a+b}{2}<b$. The 'solution' hints at adding $b$ to both sides of the inequality $a<b$, and $a+b<2b$ is as far as I've ...
4
votes
2answers
99 views

learning how to write proofs properly

I am learning how to structure my proofs in such a way that others can read them with ease. It was pointed out to me several times on this site that my proofs are not very clear. Anyway, here goes: ...
4
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1answer
148 views

For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?

If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$? Please help me to clarify the above. Thanks in advance.
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3answers
252 views

Proving or Disproving the Sum in a Set

I am doing review questions for an exam and I am completely stumped on this particular question: Let A = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}. Prove or disprove that if I select 10 distinct ...
4
votes
4answers
171 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
4
votes
1answer
714 views

Proving that a metric space is compact

Let $H^\infty$ be the set of real sequences such that each element in each sequence has $|a_n|\leq 1$. The metric is defined as $$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$ ...
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3answers
436 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
4
votes
2answers
85 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
4
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1answer
88 views

Combinatorial Identity Proof

What is a combinatorial proof for this identity: $1 \times 1! + 2 \times 2! + ... + n \times n! = (n + 1)! - 1$ I am trying to figure out what are both sides trying to count.
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2answers
338 views

Smart Pen For Math Writing

Next month I will begin to learn in the university, and I am not sure if to buy smart pen such as Livescribe or Logitech IO 1/2 to write math with.(handwriting is not an option) The problem is that I ...
4
votes
2answers
385 views

Is there a better alternative to the phrase, 'it holds that'?

The following phrases abound in my writing: There exists [whatever] such that [whatever]. For all [whatever] it holds that [whatever]. Lately, I've been feeling that the phrase 'it holds that' is ...
4
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3answers
110 views

Proof that $n \in \mathbb{N}$ by combinatorial analogue?

(Disclaimer: I'm a high school student, and my highest knowledge of mathematics is some elementary calculus. This may not be the correct terminology.) A while ago, I saw the following problem: prove, ...
4
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6answers
96 views

Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?

How can I show that $$ \sum_{i=1}^{n-1}i=\binom{n}{2}? $$ This is what I have tried, but I do not know if it is correct: Proof. Let $n=2$. Then, $$ \begin{align} \sum_{i=1}^{1}i&=1\text{, ...
4
votes
1answer
224 views

How can I prove $2\sup(S) = \sup (2S)$?

Let $S$ be a nonempty bounded subset of $\mathbb{R}$ and $T = \{2s : s \in S \}$. Show $\sup T = 2\sup S$ Proof Consider $2s = s + s \leq \sup S + \sup S = 2\sup S $. $T \subset S$ where T is ...
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4answers
98 views

Am not following algebra in a proof - what am I missing here?

So I understand the majority of the proof, but am not fully following why consequently $n^2=9a^2$. Is this because we can take our value for $n$ (which is $n=3a$) and square it, which gives us $9a^2$? ...
4
votes
1answer
127 views

prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
4
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3answers
113 views

Prove that $(g \circ f )^{-1} = f^{-1} \circ g^{-1}$

I've already proven that if we assume f is bijective and g is bijective, then $(g \circ f)$ is bijective. I've also proven that$(g \circ f)^{-1}$ exists. I'm stuck on this part, however. Any ...
4
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1answer
178 views

Found a simpler proof, now how do I know if it's original?

I've found a simpler proof for some identity/theorem, hypothetically speaking, of course ;) How do I know if it hasn't been done before? For important results it's fairly easy to find. By the way, I ...
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3answers
72 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
4
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1answer
94 views

Function Surjectivity Proof

I have this question: Prove that a function $f:X\rightarrow Y$ is surjective iff for any finite set $Z$ and any function $g:Z\rightarrow Y$ there exists a function $h:Z\rightarrow X$ such that ...
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3answers
132 views

Increasing and bounded sequence proof

Prove that the sequence $a_n= 1+ \frac 12+ \frac 13+\cdots+ \frac 1n-\ln(⁡n)$ is increasing and bounded above. Conclude that it’s convergent. This what I got so far Proof: Part 1: Proving $a_n$ is ...
4
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2answers
80 views

If $x\lt y $ for arbitrary real x and y there exists a real r $r$ such that $x \lt r \lt y$ and hence infinitely many.

If $x\lt y $ for arbitrary real $x$ and $y$ there exists a real r $r$ such that $x \lt r \lt y$ Prove that there is at least one r satisfying this inequality, and hence infinitly many. I was ...
4
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2answers
202 views

Is this proof, that $\sqrt{n}$ is irrational for all non-square $n \in \mathbb{N}$, correct or not?

Prove that the square root of all non-square numbers $n \in \mathbb{N}$ is irrational I have made an attempt to prove this, I don't know if it's correct though: Take a non-square number $n \in ...
4
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3answers
190 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
4
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1answer
75 views

The 2 Ways to Prove Uniqueness - Interchangeable or Nonidentical?

An element belonging to some prescribed set $A$ and possessing a certain property $P$ is unique if it is the only element of $A$ having property $P$. Typically, to prove that only one element of ...
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4answers
161 views

If $S_n = 1+ 2 +3 + \cdots + n$, then prove that the last digit of $S_n$ is not 2,4 7,9.

If $S_n = 1 + 2 + 3 + \cdots + n,$ then prove that the last digit of $S_n$ cannot be 2, 4, 7, or 9 for any whole number n. What I have done: *I have determined that it is supposed to be done with ...
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1answer
156 views

Hard-wiring a proof method in my head

There's s a kind of proof regularly used in linear algebra ( proving facts about Transformations, direct sums, basis, ... ) that i have definitely agreed with but still couldn't connect my intuitive ...
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1answer
112 views

Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$

I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with $$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$ First thing I want to ...
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4answers
207 views

Proof of the irrationality of $\sqrt{3}$ - logic question

Prove $\sqrt{3}$ is irrational. (Proof by contradiction). Let $\sqrt{3}$ be a rational number in simplest form $\frac pq$. So squaring both sides of $\sqrt{3}=\frac pq$ we get $3=(\frac {p}{q})^2$ ...
4
votes
2answers
371 views

Induction without integers (aka Structural Induction)

While composing the following question, I had an "ah-ha" moment. I still want to post the question along with my answer to show what I have learned. Any comments or additional answers will be greatly ...
4
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2answers
663 views

$d(x,A)=0\iff $ every neighborhood of $X$ contains a point of $A$

Mendelson, Introduction to Topology, p.52 $(8)$. Let $A$ be a non-empty subset of a metric space $(X,d)$. Let $x\in X$. Prove that $d(x,A)=0$ if, and only if, every nieghborhood $V$ of $x$ ...
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1answer
39 views

Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...
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3answers
91 views

Proving that Spec(α) and Spec(ß) partition positive integers iff α and ß are irrational and 1/α + 1/ß = 1

From Concrete Math, problem 3.13 asks: "Let α and ß be positive real numbers. Prove that Spec(α) and Spec(ß) partition positive integers if and only if α and ß are irrational and 1/α + 1/ß = 1" The ...
4
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1answer
118 views

Endomorphism- Nilpotent matrices

An endomorphism $f: V \rightarrow V$ of an $F$-vector space is called nilpotent iff there exists $ \delta \in \mathbb N$ such that $f^\delta=0$. Suppose that $f : V\rightarrow V$ is a nilpotent ...
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2answers
61 views

Proving convengent sequence theorem.

When $n$ approach to infinity prove that if $$ \lim(a_{n+1}-a_n))= 0,$$ then $a_n$ is convergent. I can prove the converse of this theorem is true but I can't prove this one. I know that since $$ ...
4
votes
3answers
256 views

Coverings of Heine-Borel.

Here is an extract from https://math.uc.edu/~halpern/calc.1/Ho/Heineborelthm.pdf Some words and sentences were cut and modified to avoid wordiness Theorem: Let $\mathcal F$ be a family of open ...