All Questions
Tagged with real-numbers analysis
217
questions
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 ...
- 467