All Questions
Tagged with real-numbers analysis
217
questions
3
votes
1
answer
177
views
Suppose $A$ is a countable subset of $\mathbb{R}$. Show that $\mathbb{R} \sim \mathbb{R} \setminus A $.
Here is the question I am trying to answer: Let $\mathbb{R}$ denote the reals and suppose A is a countable subset of $\mathbb{R}$. Show that $\mathbb{R} \sim \mathbb{R} \setminus A $.
My attempt:
...
- 159
3
votes
1
answer
109
views
Is the order inevitable in constructing the real numbers?
There are several ways to construct real numbers, such as the Dedekind cut, monotone bounded sequences and Cauchy sequences. It is obvious that the former two involves the order of the rational ...
- 4,527
3
votes
1
answer
69
views
$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $
Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above.
$ A^c = $ those element of the universe that are not in A.
$ \Bbb R =$ ...
- 1,916
3
votes
1
answer
46
views
Distance of a real number to a discrete set of scaled sine values
Let $M>0$ be an integer, $c\in(0,\frac{1}{2})$ a real number,
$$
a_{m,n}:=\frac{2n}{\pi}\sin\frac{m\pi}{2n},
$$
$$
A_n:=\left\{a_{m,n}:~m=1,\ldots,n-1\right\}, \text{ and}
$$
$$
d_n:=\operatorname{...
- 596
3
votes
1
answer
121
views
Who will win in this choosing nested intervals game? (a.k.a. Banach–Mazur game)
Question
Alice and Bob are playing a game. The rules are as follows: First Alice chooses a compact interval $A_1\subset\mathbb R$ (in this question, intervals contain more than one points, they are ...
- 13.7k
3
votes
1
answer
59
views
Extraction of pointwise convergenct subsequence using Arzela-Ascoli theorem
Let $f_n:[a, b] \rightarrow \mathbb{R}$ be a sequence of continuous functions which is uniformly bounded i.e. $||f_n||_{L^{\infty}} \leq M <\infty$ and satisfies $f_n(a)=A$ for all $n\in \mathbb{N}....
- 1,174
3
votes
1
answer
620
views
Are these valid Dedekind cuts for $e$ and $\pi$?
I took the liberty to attempt to construct Dedekind cuts for $e$ and $\pi.$ That is, come up with a set $\alpha$ of rational numbers (that would correspond to the reals $e$ and $\pi$) such that,
If $...
- 1,397
2
votes
5
answers
3k
views
Direct proof of Archimedean Property (not by contradiction)
I looked at the proof of Archimedean Property in several places and, in all of them, it is proven using the following structure (proof by contradiction), without much variation:
If $\space x \in \...
- 350
2
votes
1
answer
3k
views
Showing any real number between 0 and 1 has a unique binary expansions
Hello I have tried to prove this result as I know it is true, it is obvious but I dont know how. Ive thought of dividing the interval into two pieces and checking which side the number is on, and then ...
user172377
2
votes
2
answers
184
views
Finding an irrational number between two given irrational numbers constructively
Statement:
Let $$X=\{(a,b) \in \mathbb{R} \setminus \mathbb{Q} \times \mathbb{R} \setminus \mathbb{Q}:a<b\}$$
There exists a function $f:X \rightarrow \mathbb{R} \setminus \mathbb{Q}$ such that for ...
2
votes
3
answers
353
views
A positive real number $x$ with the property $x^3=3$ is irrational.
I have the following problems:
1) There exists a positive real number $x$ such that $x^3=3$.
2) A positive real number $x$ with the property $x^3=3$ is irrational.
My Idea for 1) would be (there ...
- 97
2
votes
3
answers
139
views
$|a-b|<\varepsilon\implies a=b$ for uniqueness of limit?
Consider the following statement:
Let $a,b\in\mathbb{R}$ be any two real numbers. Then we have that
$$\forall\varepsilon>0:|a-b|<\varepsilon\implies a=b.$$
Here is my attempt to prove it:
Proof. ...
- 77
2
votes
4
answers
743
views
How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?
Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$.
One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
- 1,893
2
votes
1
answer
671
views
Prove that the following set is dense in R
I need to show $ S = { m\cdot \sqrt{2}+ n\cdot \sqrt{3},where~m,~n~in~\mathbb{Z}} $ is dense in $\mathbb{R}$.
I showed that S has an element in $(0,ε)$ for every $ε>0.$
How do I proceed to show ...
- 163
2
votes
1
answer
135
views
Set of finite sums is dense
Let $(a_n)_{n\geq 1}$ be a sequence of non-negative real numbers such that $a_n \to 0$ but $\sum_{n\geq 1} a_n$ diverges. Show that the set of sums $\sum_{n \in S} a_n$, where $S$ ranges over the ...
- 2,813