All Questions
Tagged with real-numbers proof-writing
127
questions
2
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281
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Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
- 109
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
2
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1
answer
63
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Proving a rational number is NOT in the lower Dedekind cut for a transcendental number
I'm attempting to argue with a finitist who claims that transcendental numbers can't be defined as Dedekind cuts without using an infinite predicate or in some other way requiring an infinite number ...
- 121
2
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1
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322
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Bad Proof? Between any two reals is a rational number
I know about the proof found here: Proof there is a rational between any two reals.
I wanted to know if this similar proof is also correct?
Assume $x > 0$. Since $y > x$, it follows $y-x>0$....
2
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1
answer
214
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Proving that $ ^\mathbb{Q}\mathbb{R}$~$\mathbb{R}$ using Cantor-Bernstein [duplicate]
I am trying to prove that $ ^\mathbb{Q}\mathbb{R}$~$\mathbb{R}$ .
I want to use Cantor Schroder Bernstein Theorem rather than coming up with a bijection. Any suggestions on how to get started with ...
- 347
2
votes
1
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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
2
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3
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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
2
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0
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70
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If $0<a<b$ and $0<c<d$ then $bd+ac>bc+ad.$ Can this inequality be generalised to include more than just two lots of two numbers?
$0.7<2.4,\ $ and $\ 0.8< 0.9.$ By calculation, we see that $0.7\times 0.8 + 2.4\times 0.9 > 0.7\times 0.9 + 2.4\times 0.8.$
Indeed, for any pair of two numbers $\ 0<a<b\ $ and $\ 0<c&...
- 20.7k
1
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2
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659
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Can every real number be uniquely represented as a sum of a rational number and an irrational number $\in [0, 1)$?
I've needed to prove the transitivity of the following relation on the set of all real numbers:
$x − y$ is a rational number.
Immediately I've thought "Every real number can be uniquely ...
- 2,193
1
vote
5
answers
2k
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Proof that all real numbers have a rational Cauchy sequence?
I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known ...
- 1,681
1
vote
1
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364
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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
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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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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
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3
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How can I prove that the first three decimal digits of a real number between 0 and 1 can be equal to the reciprocal of 2 to the power of that number?
This is the challenge problem at the end of Chapter 1 of Solow's How to Read and Do Proofs.
The problem:
Find a counter-example to the following statement: “If x is a
positive real number ...
- 177
1
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1
answer
63
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When $ a^5 < 5 $ show that there exists b such that $ a<b, b^5<5 $
Here's my approach.
Since $ a^5 < 5, a<\sqrt[5]{5} $
By density of rational number, there must be integer $m$ and natural number $n$ such that
$ a< \frac{m}{n} < \sqrt[5]{5}$
If I let $...
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