All Questions
Tagged with real-numbers proof-writing
127
questions
1
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2
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216
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Show that rational cosets are either identical or disjoint
Let $\mathbb{Q}$ denote the set of rational numbers.
Let $x,y \in \mathbb{R}$. Let $A_x = x+ \mathbb{Q} , A_y = y+ \mathbb{Q} $
Can someone help me in simple arguments prove that cosets $A_x, A_y$ ...
1
vote
1
answer
27
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Conditions required for Inequality to hold
Given that $0 \leq X \leq x \leq y \leq Y < \infty$, I am interested to know the condition whereby $\frac{\sqrt{y}-\sqrt{x}}{\sqrt{y}+\sqrt{x}} \leq \frac{\sqrt{Y}-\sqrt{X}}{\sqrt{Y}+\sqrt{X}} $ ...
- 13
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votes
2
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99
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Cardinality proof verification
Problem:
Let $C \subset (0,1)$ be the set of all numbers whose unique decimal representation contains the number seven. Show that the number of elements in $C$ must be the same as the number of ...
- 422
2
votes
1
answer
434
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How to construct binary sequences associated to points of the Cantor set?
Let $C\subset \mathbb{R}$ be the Cantor set, obtained from the interval $[0,1]\subset \mathbb{R}$ by removing the middle thirds of successive subintervals. That is, assuming $C_n$ constructed we let $...
- 26.9k
1
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1
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365
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(Ir)rationality of Real Numbers
I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right?
For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational
...
- 59
1
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4
answers
8k
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Prove $(-x)y=-(xy)$ using axioms of real numbers
Working on proof writing, and I need to prove
$$(-x)y=-(xy)$$
using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...
user124862
1
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1
answer
54
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My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$
Is it reasonable to prove the following (trivial) theorem?
If yes, is there a better way to do it?
Let $x, y \in \mathbb{R}$.
Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$.
$\textbf{...
- 399
0
votes
1
answer
104
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$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)
As an exercise, I tried to prove the following theorem.
Please share your thoughts about what I wrote.
(The proof only uses the utensils which are listed below.)
Theorem
Let $x,y,z \in \mathbb{R}$.
...
- 1,321
3
votes
1
answer
2k
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$x^n y^n = (xy)^n$, proof exercise
As an exercise, I tried to prove the following theorem.
Please share your thoughts about what I wrote.
(The proof only uses the utensils which are listed below.)
Theorem
\begin{equation*}
x^n y^n ...
- 1,321
3
votes
2
answers
1k
views
$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon>0$. How to prove $x \le y$? [closed]
$x,y \in \mathbb R$ are such that $x \lt y + \epsilon$ for any $\epsilon \gt 0$ Then prove $x \le y$.
1
vote
4
answers
120
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Proof: For every $x \in \mathbb R\,\exists n\in\mathbb Z,\,r\in[0,1):x=n+r.$
I still do not understand how to approach proofs. Any help would be appreciated.
For every real number $x$, there exists an integer $n$ and a real number $r\in [0,1)$ such that $x = n + r$.
Hint: ...
- 35
1
vote
2
answers
4k
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An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$
Problem:
Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon &...
- 6,684
2
votes
3
answers
75
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If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof
For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).
Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.
Assume, x, y ∈ ℝ # Domain assumption
...
- 93
1
vote
1
answer
2k
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There exists a positive real number $u$ such that $u^3 = 3$
Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$.
Show that there exists a positive real number $u$ such that $u^3 = 3$.
So far, I have come up with the ...
- 81
0
votes
1
answer
236
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Identify a countable union of nested intervals using the Archimedean principle [closed]
$\displaystyle\bigcup_{n=2}^\infty \left[\frac1n,3-\frac 2n\right]=(0,3)$. I can't prove using limits. I have to use the Archimedean principle and I don't know how to go about doing that..
- 41
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3
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85
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$a > b+1 \Rightarrow a>x>b$?
If I have $a,b \in \mathbb R$ such that $$a > b+1 $$
It is assured that $\exists\space x \in \mathbb Z: a>x>b$
Does this property have some special name?
How can this be proved?
This idea ...
- 6,834
0
votes
1
answer
43
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Proving of Inequalities
How to prove:
If $a>0$ and $b>0$, and $a^2>b^2$, then $a>b.$
I've tried different methods but I really can't prove this one. Thank you for your help!
2
votes
2
answers
704
views
How to show that an infinite decimal is equal to a unique real number?
I don't understand how the proof above shows that two distinct real numbers correspond to different infinite decimal.
All I got out of the explanation is given any two distinct real numbers $a$ and $...
- 3,600
0
votes
1
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73
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Help with 2 questions my professor gave us
I was wondering how to solve these two proofs my professor put on the blackboard today. He said they were pretty easy but i'm still unsure how to prove them. ANy help would be greatly appreciated!
(i)...
- 107
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3
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72
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Help solving a proof
My professor put this up on the blackboard and I was wondering how to solve it.
Let $x,y \in \mathbb{R}$. Then |$x$|< |$y$| if and only if $x^2 < y^2$.
- 107
0
votes
3
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381
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Having trouble understanding Cantors proof that real numbers are uncountable
I found this video very easy to follow and understood the proof.
https://www.youtube.com/watch?v=mEEM_dLWY0g
However, I am still having trouble understanding the proof presented to me in my csmath ...
- 25
2
votes
1
answer
398
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My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.
What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Among three elements of ...
- 1,321
3
votes
0
answers
190
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My first simple direct proof (very simple theorem on real numbers). Please mark/grade.
What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Let $I = [a,b]$ be a non-empty closed ...
- 1,321
1
vote
1
answer
125
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Possible book correction or am I missing something?
Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
- 825
3
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3
answers
467
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Formal proof of: $x>y$ and $b>0$ implies $bx>by$?
Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$.
What is a formal (low-level) proof of this result? Or is this property taken as axiomatic?
The motivation for this ...
0
votes
1
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82
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Math proof involving open and closed intervals
So this is a multipart question, and I have a couple "theories" on how to attack it but still stuck in the infancy stages. Basically, not very far.
For each real $r>0$ let $I_r$ denote the open ...
- 85
5
votes
1
answer
6k
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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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