All Questions
Tagged with real-numbers analysis
217
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
892
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"...
- 477
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 $k\gt 0$ ...
- 5,601
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
312
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 ...
- 133
4
votes
2
answers
4k
views
Show $(a, b) \sim\Bbb R$ for any interval $(a, b)$ [duplicate]
Show $(a, b)$ has the same cardinality as $\Bbb R$ for any interval $(a, b)$.
From the previous chapters of the book (Understanding Analysis by Stephen Abbott), I know that $(-1,1)$ has the same ...
- 1,689
4
votes
1
answer
197
views
Is there any construction of real numbers that does not use a quotient space in the process?
I am working on an isomorphism (in terms of order and operations) from the power set of integers to R. I would like to know if anyone knows of any construction of the real numbers that uses such a ...
- 181
4
votes
2
answers
99
views
Solve the integral? [closed]
Help me? please How to solve this integral?
$$\int\frac{1+x^2}{1+x^4}\,dx$$
- 41
4
votes
1
answer
177
views
Abstract concept tying real numbers to elementary functions?
Real numbers can be broken into two categories: rational vs. irrational. Irrational numbers can be approximated, but never fully represented by rational numbers.
Analytic functions have Taylor ...
- 247
4
votes
1
answer
846
views
Baby Rudin Problem Chapter 2, Problems 17(c) and (d)
Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Then I've managed to show that (a) $E$ is not countable, and (b) $E$ is not dense in $[0,1]$.
...
- 26.9k
4
votes
1
answer
191
views
Pick out true statements about the limit of $f_n(x)=\frac{1}{1+n^2x^2}$
For the sequence of functions $f_n(x)=\frac{1}{1+n^2x^2}$ for $n \in \mathbb{N}, x \in \mathbb{R}$ which of the following are true?
(A) $f_n$ converges point-wise to a continuous function on $[0,...
- 195
3
votes
4
answers
181
views
Does $\{a:a=bc\text{ for }b,c\in\mathbb{Q},\text{ and }b\le t,\text{ }c\le u\}$ contain all rationals $a\le tu?$
Note that $t,u\in\mathbb{R},$ and $t,u>0.$ What if, for some $a\le tu,$ all possible candidates $b,c$ reside in outer scope? If the above statement (in the title) is true, the proof must be the ...
3
votes
2
answers
160
views
Is the function $\,f(x, y) = x-y\,$ closed?
Is the function $\,\,f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$, defined as $$f(x,y)=x-y,$$ closed?
3
votes
3
answers
5k
views
Is supremum part of the set or it is the bigest element out of it?
"If $\sup A$ is in $A$, then is $\sup A$ called also Maximum: $\max A$."
So that means that $\sup A$ can be outside the set $A$?
And lastly the upper barrier(or bound, not sure) is from $A$, right?
3
votes
1
answer
280
views
Why can we not write the reals as a countable union of sets
I understand that the reals are not countable, what goes wrong with this? $$\mathbb{R} = \bigcup_{n=1}^\infty (-n,n) $$
- 77
3
votes
2
answers
95
views
Is $\mathbb{R}\cup\{-\infty,+\infty\}$ the categorical co-completion of $\mathbb{Q}$
Seeing $\mathbb{Q}$ as an ordered set, the colimit of a diagram $D:\mathcal{I} \to \mathbb{Q}$, when it exists, is just $\operatorname{colim}D \cong \operatorname{sup}_iD(i)$.
It seems to me that ...
- 33
3
votes
3
answers
1k
views
For all real numbers satisfying $a < b$, there exists an $n \in \mathbb{N}$ such that $a + 1/n < b.$
From Stephen Abbott's Understanding Analysis 1.2.11:
For all real numbers satisfying $a < b$, there exists an $n \in \mathbb{N}$ such that $a + 1/n < b.$
My try:
$$\forall a\in \Bbb R, \forall ...
- 71
3
votes
1
answer
89
views
What is the difference between $\mathbb{R}$ and $\mathbb{R}^1$?
I am wondering that $\mathbb{R}$ and $\mathbb{R}^1$ are same or not. $\mathbb{R}$ is the real numbers, and $\mathbb{R}^1$ is a set of 1-tuples. I am so stucked on this. Thanks for the support.
- 35
3
votes
3
answers
295
views
Are there good lower bounds for the partial sums of the series $\sum 1/\log(n)$?
Consider the partial sums $$S_n = \sum_{k=2}^n\frac{1}{\log(k)}.$$ Are there good lower bounds for $S_n$ as $n\to\infty$ ?
I am not necessarily looking for sharp bounds (although they would be nice), ...
- 2,044
3
votes
1
answer
241
views
Show that $f(x) = \sum_{k=0}^{\infty}{\frac{\sin^2(kx)}{1+k^2 x^2}}$ uniformly converges.
I want to show that $f(x) = \sum_{k=0}^{\infty}{\frac{\sin^2(kx)}{1+k^2 x^2}}$ uniformly converges for $|x| \geq \delta$ for any given $\delta > 0$.
I don't know how to use the M-test here, since ...
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