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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719 views

What is the relation A = B = C called in a proof?

When writing a proof if I have the relationship $$ A = B = C $$ And I want to use that to prove $$ A = C $$ I remember there being some term for it. What is that term, and what would be an ...
4
votes
5answers
428 views

Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$

Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$ I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!
4
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2answers
319 views

Proof that the Irrationals are Countable

Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs ...
4
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4answers
175 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...
4
votes
4answers
271 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
4
votes
2answers
213 views

Proof of $f(\{x\})=\{f(x)\}$

I thought, is this really that simple? Or am I missing a piece? This is my proof: $f(\{x\})=\{f(x)\}$. Look at $f(\{x\})$. By definition, $f(\{x\})=\{f(a)|a \in \{x\}\}$, and therefore ...
4
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2answers
2k views

A continuous function $f$ from a closed bounded interval $[a, b]$ into $\mathbb{R}$ is uniformly continuous

I am reading (as a supplement) the book Basic Real Analysis, by Anthony Knapp. Before I proceed into reading a proof, I want to be sure that the result seems obvious. Yet, I am having trouble seeing ...
4
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5answers
337 views

Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: ...
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2answers
1k views

Prove that such an inverse is unique

Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$ $z^{-1}$ is also often written as $1/z$
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2answers
147 views

Prove that: $(1+a_1)(1+a_2)…(1+a_n) \le 1 + S_n + (S_n)^2/2! + … + (S_n)^n/n!$

I am currently working on this problem from Hardy's Course of Pure Mathematics and have gotten stuck near the end. I was wondering if someone could help me determine what to go next. Question If ...
4
votes
2answers
813 views

Showing there are no integer solution to equation $\;2^x = 4y+3$

I am stuck on this problem and I'm not sure how to approach it. Can anyone help me out with figuring how to approach the proof? My task is to: Prove that it is impossible to find integers $\,x,\, ...
4
votes
1answer
707 views

are two consecutive numbers relatively prime?

I have a question. I have been given this proof: "For any $n$ in the integers where $n>2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$." I have gone through and actually ...
4
votes
2answers
656 views

Prove that $x^{2} \equiv -1$ (mod $p$) has no solutions if prime $p \equiv 3\pmod 4$.

Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$ Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$. I know this problem has something to do with Fermat's Little ...
4
votes
3answers
2k views

Proving the so-called “Well Ordering Principle”

Is there anything wrong with the following proof? Theorem. Every non-null subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
4
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5answers
258 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
4
votes
3answers
228 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
113 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
votes
3answers
97 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 ...
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4answers
150 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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votes
1answer
14k 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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3answers
39 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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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
168 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
votes
1answer
171 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
votes
3answers
312 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: ...
4
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3answers
376 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
809 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
158 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 ...
4
votes
4answers
200 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
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2answers
117 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
votes
1answer
156 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.
4
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3answers
260 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
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4answers
177 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
3answers
1k views

Proving that the reciprocal of an irrational is irrational

The question I am working on is: Prove that if x is irrational, then 1/x is irrational. My proof differs from the one given in the answer key; but I still feel that mine is valid. Could someone ...
4
votes
1answer
750 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 $$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$ ...
4
votes
3answers
488 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 ...
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votes
4answers
79 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
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2answers
106 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 ...
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1answer
93 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
419 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
614 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
111 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
votes
6answers
97 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{, ...
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1answer
243 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 ...
4
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2answers
251 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
4
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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
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3answers
117 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
votes
1answer
182 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 ...
4
votes
1answer
40 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...