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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2
votes
2answers
30 views

Prove that Set B is countable - Is this proof correct?

It seems that I have some issues with the rigor of this proof and I don't know what I'm doing wrong. Could someone tell me if this proof is correct and rigorous enough? Here's the question Prove ...
3
votes
1answer
38 views

Show that $\bar A = A \cup [(0,0), (0,1)]$

In $(\mathbb R^2, ||\cdot||_{\infty})$, let: $A_0 = ]0,1] \times \{0\}$ $A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$. $A = \cup_{n=0}^{\infty} A_n$ It is required to prove that: ...
0
votes
1answer
31 views

Extending the transitive property [on hold]

Suppose we have a transitive relation $R$ on a set $S$. Suppose for some $n\in\mathbb{Z}^+\colon (s_0, s_1),(s_1,s_2),\ldots,(s_{n-1}, s_n)\in R$. Show that: $(s_0, s_n) \in R$ So I am having ...
0
votes
1answer
36 views

what's the answer for this proof [on hold]

What's the answer for this question: Show that: a) a−∅ = a. b) ∅−a = ∅. I am already try to solve this, but I feel this is not logical solution. a) A−∅ = A A−∅ ={x|x ∈A^ ...
1
vote
4answers
60 views

Solution verification: Prove by induction that $a_1 = \sqrt{2} , a_{n+1} = \sqrt{2 + a_n} $ is increasing and bounded by $2$

I have the following recursive relation (sequence): \begin{align} a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 + a_n} \end{align} My Try: I'm a little skeptical of my manipulations near the end but it ...
2
votes
1answer
41 views

Proving an Inequality (terms won't cancel out)

Problem: If $x$ and $y$ are real numbers such that $y \geq 0$ and $y(y+1) \leq (x+1)^2$, prove that $y(y-1) \leq x^2$. This is what I tried: \begin{align} y(y+1) \leq (x+1)^2 &\implies y^2 + y ...
2
votes
0answers
20 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
0
votes
0answers
34 views

How to prove this set P is countable? [duplicate]

Hi so I'm a beginner to proofs and these day's I'm studying infinite sets. I'm trying to figure out the proof for the following: Let P = {X$\in \mathscr{P}({\mathbb{Z}}^+)$| X is finite}. Prove ...
1
vote
0answers
38 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
1
vote
0answers
38 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
1
vote
1answer
52 views

What should I learn to increase my skill to find proof?

I know... reading lot of proofs and comments about them and working hard by myself on proving theorems are probably the only good solutions. But in the same time, it is not a solution at all because ...
1
vote
4answers
66 views

Prove that for all positive integers $n$, $2^1+2^2+2^3+…+2^n=2^{n+1}-2$ [duplicate]

I want to prove that for all positive integers $n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$. By mathematical induction: 1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$ 2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, ...
0
votes
3answers
42 views

How to show $f(x) = x^2 + x + 1$ is continuous? [on hold]

Using the $\epsilon-\delta$ definition of continuity to prove that a particular function is continuous: Let $f(x) = x^2 + x + 1$. Given a positive number $\epsilon$, find a positive $\delta$ such that ...
0
votes
2answers
56 views

Explaining why proof by induction works [duplicate]

I am learning math proofs for the first time. So I cannot discern the reason for all the details in a proof. Here's the statement of mathematical induction: For every positive integer $n$, let ...
1
vote
2answers
25 views

Proof by induction of the Inequality of Harmonic numbers: $H_{2^n} \ge 1+ \frac n2$

My question is, for the question below, in the inductive step, where does $\dfrac{1}{2^{(k+1)}}$ come from?And where does $2^k$ come from in the third last step?
3
votes
1answer
27 views

Uniqueness Proof procedure

I'm reading a book on understanding math proofs to enable me to understand mathematics at a deeper level. Along the way I came across this: An element belonging to some prescribed set $A$ and ...
0
votes
1answer
39 views

I am trying to use proof of sequence correctly to make valid

Here I am trying to use a proof sequence so that the argument is valid (hint: the last A’ has to be inferred). (A → C) ∧ (C → B') ∧ B → A' Here are my steps I tried but not sure if this is correct ...
0
votes
0answers
40 views

Trying to justify each step correctly in proof sequence

I am trying Justify each step in the proof sequence below for correctly with [A → (B ∨ C)] ∧ B' ∧ C' → A' So I justified my steps here but I am not sure at 1 to 3 if I did it correctly. A → (B ∨ ...
2
votes
4answers
386 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
3
votes
1answer
46 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
0
votes
2answers
46 views

Power set equinumerosity. Is this proof correct?

So I'm trying to prove the following, Prove that if $A\sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$. Here's how I started out to prove there is a function that is injective: Suppose $A ...
4
votes
1answer
40 views

Introduction to proofs. [duplicate]

I am not at all familiar with mathematical proof-writing and would like to learn how to create my own proofs. So, I was wondering whether it would be possible for you to recommend me to any book or ...
1
vote
2answers
41 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
-2
votes
0answers
50 views

Summation Direct Proof Help [on hold]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
0
votes
1answer
37 views

How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
1
vote
2answers
38 views

Proving $10^n \equiv 1 \pmod 3$ for all $n\geq 1$ by induction

Prove that $10^n \equiv 1 \pmod 3$ for all positive integers $n$ by mathematical induction. Can someone please help me in solving this problem and explain what's going on? Any guidance would be ...
1
vote
1answer
39 views

Formal language: Proving the reverse operation on a word through induction

I'm practicing proofs and given the following statement: Let $\Sigma$ be an alphabet, $\epsilon$ the empty word and $\sigma:\Sigma^{*}\rightarrow\Sigma^{*}$ an operation which for $a\in\Sigma$ and ...
-2
votes
0answers
38 views

Why does this proof by bashing not work?

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. The feet of the perpendiculars from $C$, $B$, and $A$ to the opposite sides are $F$, $E$, and $D$ respectively. Prove that ...
1
vote
1answer
59 views

Proving that the gamma function is a certain limit

This time I want to prove that $\displaystyle \Gamma(x) = \lim_{\varepsilon\rightarrow 0} \int_\varepsilon ^{1/\varepsilon} t^{x-1} e^{-t}$, I know this is true because we have defined $\displaystyle ...
0
votes
1answer
14 views

Using existential instantiation on a universally quantified given

I'm trying to prove the following exercise of How to Prove it: A structured Approach (Section 3.4, exercise 19): Suppose A, B and C are sets. Prove that A $\triangle$ B and C are disjoint iff A ...
1
vote
0answers
18 views

Is there a standard notation for $(p_i-k)(p_{i-1}-k)(p_{i-2}-k)\cdots$ where $k$ is a small positive integer

For $k=0$, there is: $p_i\# = (p_i)(p_{i-1})(p_{i-2})\cdots(5)(3)(2)$ For $k=1$, there is: $\varphi(p_i\#) = (p_i-1)(p_{i-1}-1)(p_{i-2}-1)\cdots(5-1)(3-1)(2-1)$ Is there any other notation that ...
3
votes
1answer
16 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
2
votes
0answers
19 views

Proof: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$

Prove: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$ This is my work so far: Let P be any point of the plane and set: $P'=T_{AB} (P)$ We want to show ...
0
votes
2answers
44 views

Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
0
votes
0answers
54 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
0
votes
0answers
12 views

Upper bounds and Lower bounds (Relations Proof Problem)

So I've only recently started studying proofs and I've been using Velleman's "How to Prove it" This is a theorem from the book. I'm having a hard time on proving it. Suppose A is a ...
0
votes
0answers
35 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...
0
votes
1answer
25 views

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of $\mathbb{R}$.

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of R. I know that I need to show that the complement of this set is open in order to show that this set is closed. The ...
0
votes
2answers
51 views

Prove or disprove: There exists a prime p > 3 such that p + 2 and p + 4 are also prime

I'm having a lot of difficulties with this proof. Can someone please solve it and explain to me what's going on at each step? Thank you!
1
vote
2answers
32 views

Proof by Elements to Show $D^{c} ⊆ A^{c}$

Use proof by elements to verify that for all nonempty sets $A$, $B$, and $D$ if $A ⊆ B$, $D^{c} ⊆ B^{c}$, then $D^{c} ⊆ A^{c}$. Here's the proof I have written so far. I have gotten feedback that ...
2
votes
2answers
250 views

how to prove that the following is not a regular language?

the language we want to disprove is : $$ L = \{ 0^i1^j| gcd(i,j)=1 \} $$ my attempt : i used the pumping lemma this way: consider the set of strings of the form $0^p1^q$ such that $n <=p$ and ...
4
votes
2answers
94 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
0
votes
1answer
25 views

Let S ⊂ R 2 . Show that if cl(S) is convex, then S is convex.

Let $S ⊂ R^2$. Show that if $cl(S)$ is convex, then S is convex. This seems intuitive but, I am having trouble thinking of a proof or counter example.
0
votes
0answers
53 views

Proving Integrability of $sgn(\sin(\frac{\pi}{x}))$

I must show that for $f(x) = sgn(\sin(\frac{\pi}{x}))$ on $[0,1]$, that $f$ is Integrable. I know that a function is integrable if the Upper and Lower sums of $f$ coincide. That is, if $$U(P,f) - ...
0
votes
2answers
29 views

Proving that the set of languages over an alphabet Σ is a monoid regarding concatenation

I'm practicing proofs and would like to prove that the set of languages over an alphabet $\Sigma$ is a monoid regarding concatenation by showing that the following statements are true: There is a ...
0
votes
2answers
152 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
1
vote
4answers
74 views

Suppose that $F'(x)\leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x)\leq G(x)$ for all $x \in \mathbb{R}$.

Suppose that $F'(x) \leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x) \leq G(x)$ for all $x \in \mathbb{R}$. Prove or disprove. I came up with counterexample that $\cos(x) \leq \cos(x)+1 $ for all ...
1
vote
3answers
44 views

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length.

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length. Prove or disprove. I got the idea that they are inverse functions and probably we can show ...
1
vote
3answers
33 views

How to prove that $G(x)=ax+b$ is a one-to-one correspondence where $a\neq0$ and $a,b\in\mathbb{R}$.

$G(x) = ax + b$ where $a$ is not equal to $0$ and $b$ are real numbers. Prove $G$ is a one-to-one correspondence. I understand that for every $a$ there is a corresponding $b$-value that does not ...
1
vote
3answers
60 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...