All Questions
Tagged with real-numbers proof-writing
127
questions
0
votes
1
answer
67
views
Trouble proving the floor function is onto with the domain being all real numbers
I need to prove that for the mapping $f : \mathbb{R} \mapsto \mathbb{Z} $ given by $ f(x) = \lfloor x \rfloor$, $f$ is onto. I know how I would do it if both the domain and codomain were both $\mathbb{...
0
votes
2
answers
1k
views
How to show/proof that the union of two non empty subsets of ${\Bbb R_{}}$ has a least upper bound?
We have two sets ${E}$ and ${T}$, that are non empty subsets of ${\Bbb R_{}}$ and are bounded above.
How can I prove that,
${E}$ ${\cup}$ ${T}$ has a least upper bound (supremum), and that ${\sup(E\...
- 414
0
votes
2
answers
2k
views
How can I proof the infimum and supremum of this set?
$E = \{{x+y : x,y \in\Bbb R_{>0}}$}
I was able to figure out that this set does not have a supremum, but I am not able to prove it. Also, how can I prove the infimum of this set ?
This is my ...
- 414
0
votes
1
answer
896
views
How to prove comparability property & writing its proof
I am given the relation in $\mathbb{R}$: $xRy$ if $x\le 2^y$. I want to prove this has the comparability property, so I know I start with let $x,y\in \mathbb{R}$. Then I need to show either $xRy$ or $...
- 1,088
0
votes
2
answers
384
views
Proving some properties of real numbers using predicate logic
I am trying to understand how some basic properties of the real numbers can be proved from axioms expressed in predicate logic.
I start by accepting the field axioms of real numbers, in addition to ...
- 777
1
vote
2
answers
1k
views
Help determining if a finite subset of $\mathbb R$ is closed and bounded.
If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
- 329
3
votes
2
answers
3k
views
How to prove the power set of the rationals is uncountable?
Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
- 7,252
0
votes
1
answer
407
views
Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.
As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations.
...
- 1
0
votes
0
answers
20
views
Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
I am trying to formally prove:
$ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
where n is an integer, and x and y are natural numbers.
It is obvious that, when $\frac xy$ is ...
- 101
7
votes
5
answers
2k
views
Is this direct proof of an inequality wrong?
My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
- 81
1
vote
2
answers
216
views
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
views
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
0
votes
2
answers
99
views
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
views
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
vote
1
answer
364
views
(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