All Questions
Tagged with real-numbers analysis
217
questions
2
votes
1
answer
100
views
Find all limit points of the sequence $\frac{1}{2}; \frac{1}{3}; \frac{2}{3}; \frac{1}{4}; \frac{2}{4}; \frac{3}{4}; \frac{1}{5}; \frac{2}{5}; \dotso$
I know that the answer is all real numbers on $[0, 1]$, but I can’t pick out the subsequences.
Please help. Thanks in advance for your time.
- 35
2
votes
1
answer
1k
views
Prove that $\mathbb{Q}\!\smallsetminus\!\mathbb{Z}$ is dense in $\mathbb{R}$
Can someone just tell me if this is a correct way to prove it.
let $(a,b)$ be a nonempty open interval in $\mathbb{R}$. Then by density of $\mathbb{Q}$ in $\mathbb{R}$ there exists $q\in \mathbb{Q}$ ...
- 1,419
2
votes
1
answer
281
views
Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
- 109
2
votes
1
answer
39
views
Show that $2-2e^{-|x|}\leq C|x|^{r}$ for some constant $C, r>0$.
I am in the middle of a proof, and I need to show that
For $x\in\mathbb{R}$, we have $2-2e^{-|x|}\leq C|x|^{r}$ for some constant $C, r>0$.
The claim can surely be reduced to show that $1-e^{-...
- 5,930
2
votes
1
answer
112
views
Cauchy sequences always have a largest or smallest element past an arbitrary index
The problem is as follows:
Let $(x_n)_n$ be Cauchy. Show that either
$$
(1)\; \forall N\in\mathbb{N}, \exists\bar{n}\geq N \text{ s.t. } \forall n\geq N, x_n\leq x_{\bar{n}}
$$
or
$$
(2)\; \forall ...
- 391
2
votes
1
answer
87
views
Where is this function continuous/differentiable?
Let $f: \mathbb R \to \mathbb R$ defined by $$f(x):= \begin{cases} x\sin x & x \in \mathbb Q \\ 0 & x\in \mathbb R \setminus \mathbb Q \end{cases}$$
In which $x\in \mathbb R$ is $f$ continuous ...
- 2,825
2
votes
1
answer
6k
views
Given a number ε > 0, prove there exists a natural number $N$ such that 1/N < ε
I believe there are three cases. I think I have figured out the first one.
Case 1: Let $ε$ = 1 and N > 1. Take $ N$=2. Then 1/2 < $ε$ ≡ 1/2 < 1.
Case 2: Let 0<$ε$<1. I believe that I ...
- 527
2
votes
1
answer
100
views
$\forall \varepsilon \in \mathbb{R}_{>0} \exists v \in V$ so that $x - \varepsilon < v$ if and only if $x = \sup V$
I'm trying to prove that, given a non-empty bounded set $V$ so that $V \subset \mathbb{R}$ that an upperbound
$x \in \mathbb{R}$ is a supremum of V if and only if $\forall \varepsilon \in \mathbb{R}_{&...
- 559
2
votes
1
answer
1k
views
Is a real closed, bounded interval a locally compact Hausdorff space?
Does this hold? I've been confused by the statement of the Riesz-Markov-Kakutani representation theorem; that is, the formulation is as follows:
Let $X$ be a locally compact Hausdorff space. For ...
- 613
2
votes
2
answers
107
views
Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb R-S\}=0$?
I founded the following question a good challenge in real analysis and topological properties of real line...
Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb R-S\}=...
- 5,466
2
votes
1
answer
80
views
Is this following inequality with increasing powers of the components true for small $x$? And if yes, what's the positive constant $C?$
Let $0< k_1\le k_2\dots \le k_m, k_i \in \mathbb{N}, k_1 \text{ an even positive integer }. f(x_1\dots x_m):=\sum_{i=1}^{m}{x_i}^{k_i}.$ I wanted to prove, if possible, that in a small enough ball $...
- 2,364
2
votes
1
answer
64
views
A question about multiplication of Dedekind cuts
If $a,b\in \mathbf{Q}^+$, how to prove $\{p\ |\ p\in \mathbf{Q}, 0<p<ab\}\subset \{rs\ |\ r,s\in \mathbf{Q},\ 0<r<a,\ 0<s<b\} $?
Here is my try:
Suppose $q\in \{p\ |\ p\in \mathbf{Q},...
- 174
2
votes
1
answer
313
views
A question about Dedekind cut in Rudin's Principles of Mathematical Analysis
In this Rudin's book, he introduces Dedekind cut at the end of chapter 1. I provide the context first. Then I describe my question carefully.
$\ $
$\ $
In the first picture, it shows the "field ...
- 174
2
votes
1
answer
83
views
Existence of a certain decreasing function
I'm in the following strange situation:
$\alpha:[0,\infty)\to\mathbb{R}_{>0}$ is a continuous, positive, decreasing function with $\int\alpha=\infty$. Let $f$ be the function $x \mapsto e^{-2x}.$
...
- 1,589
2
votes
1
answer
267
views
Trying to understand existence theorem on Rudin's Principles
I'm reading Rudin's Principles of Mathematical Analysis. It is said in the book, 1.19 Theorem (pg-8): "there exists an ordered field $\mathbb{R}$ which has the least-upper-bound property. Moreover, $\...
- 39