All Questions
Tagged with real-numbers analysis
216
questions
49
votes
7
answers
8k
views
Is the real number structure unique?
For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.
In my analysis class, our book ...
- 2,233
31
votes
6
answers
4k
views
Are there many fewer rational numbers than reals?
Today my professor asked me to figure out the probability of getting a rational number from $[0,1]$.
His answer was that the probability is $0$.
Why is this?
- 605
25
votes
5
answers
2k
views
Were "real numbers" used before things like Dedekind cuts, Cauchy sequences, etc. appeared?
Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers.
I'm also very interested, if the answer is "...
- 13.5k
8
votes
2
answers
322
views
Opsure and Clinterior of subsets of $\mathbb{R}$
We are considering the made up words opsure and clinterior, where the
opsure of a set $A$ is the smallest open set containing $A$ and the clinterior of a set $B$ is the largest closed set contained in ...
- 557
8
votes
3
answers
7k
views
When can a set have an upper bound but no least upper bound?
So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'.
I don't understand how a ...
- 341
6
votes
4
answers
887
views
How can I show the incompleteness of the irrational numbers?
To show the incompleteness of the rational numbers, we just had to find a set of rational numbers, that does not have an supremum / infimum which is element of $\mathbb{Q}$. For example, we could show ...
- 884
6
votes
2
answers
3k
views
can I find the closest rational to any given real, if I assume that the denominator is not larger than some fixed n
Given $n\in\mathbb{N}$ and $r\in\mathbb{R}$(or $r\in\mathbb{Q}$), is that possible to find a rational $\frac{a}{b}$such that, $b<n$ and $\frac{a}{b}$ is the "closest" rational to r?
With "closest"...
- 377
6
votes
2
answers
1k
views
Defining $N$ in the $\epsilon$-$N$ definiton of convergence
In my Real Analysis class we've been spending some time talking about the $\epsilon$-$N$ definition of convergence. The book we are using, Elementary Analysis by Ross, defines convergence as:
A ...
- 331
6
votes
1
answer
2k
views
approximate irrational numbers by rational numbers
I want to prove this below:
(1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
- 1,519
6
votes
2
answers
1k
views
Prove Lagrange's Identity without induction
Prove Lagrange's Identity without induction.
$$
\sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2
$$
I ...
- 13.5k
5
votes
1
answer
1k
views
Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma
Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
- 5,591
5
votes
1
answer
117
views
The existence of a measurable set with measure between rationals and the reals [duplicate]
Is there a measurable subset E ⊆ R such that whenever a < b are real numbers we have both $m(E ∩ (a, b)) > 0$ and $m((a, b) -E) > 0$ ?
This is an extra question on my real analysis class, ...
- 855
4
votes
1
answer
282
views
Is the range of an injective function dense somewhere?
Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? ...
- 2,397
4
votes
2
answers
310
views
Points of continuity of the function $f(t)=\frac{p+\sqrt{2}}{q+\sqrt{2}}- \frac{p}{q}$.
Let $f$ be the function $f(t)=\frac{p+\sqrt{2}}{q+\sqrt{2}}- \frac{p}{q}$ if $t \in \mathbb{Q}$ and $t=\frac{p}{q}$ with $p,q$ relatively prime. $f(t)=0$ if $t \not\in \mathbb{Q}$.
At which points is ...
- 2,737
4
votes
4
answers
164
views
Is there any way to prove that $\sqrt {n-1} + \sqrt n + \sqrt {n+1}$ is irrational? [closed]
Before this is marked as a duplicate I just want to say that I've already read a similar thread, where the original poster asked how they would prove that $\sqrt 2 + \sqrt 5 + \sqrt 7$ is an ...
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