All Questions
26
questions
6
votes
5
answers
5k
views
What is a natural number? [duplicate]
According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the ...
- 30.2k
1
vote
4
answers
1k
views
If $b-a>1$ then there is a $k\in \mathbb{Z}$ such that $a<k<b$
Given $a, b \in \mathbb{R}$, such that $b-a>1$, there is at least one $k\in \mathbb{Z}$ such that $a<k<b$.
My attempt:
Consider $E:=(a,b)\cap \mathbb{N}$. We need to show that $E$ is not ...
0
votes
1
answer
26
views
On pairs of integers satisfying an inequality
Let us consider the following set $A:=\{(r,t)| r \in \mathbb N \cup \{0\}, t \in \mathbb Z, t \leq r-5\}$
My question is the follow : Does there exists any pair $(r,t)$ belonging to $A$ such that it ...
- 15.5k
7
votes
2
answers
4k
views
Let $x$ be a real number. Prove the existence of a unique integer $a$ such that $a \leq x < a+1$
Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$
...
- 547
3
votes
1
answer
89
views
Irrationality of an "Euler-like" number
Let $(a_n)_{n=0}^{\infty}$ be a sequence of zeroes and ones such that $a_n=1$ for infinitely many $n$. Let $\displaystyle x:=\sum_{n=0}^{\infty} \frac{a_n}{n!} .$
Is $x$ irrational? I believe it is, ...
- 511
0
votes
0
answers
44
views
Is this proof missing a "push-up" step? Every non-empty set of real number bounded from below has a greatest lower bound.
This is from BBFSK, Vol-I pages 142 and 143. The reference to $\S{4.1}$ means for whole numbers $a_i\lt{g}$ and $g\gt{1}$
$$r_n=\sum _{i=0}^{n} a_i g^{-i}.$$
The expression in the sentence following (...
- 3,349
1
vote
1
answer
62
views
Can we guarantee that there exists an $\epsilon' > 0$ such that holds for this inequality?
I am currently trying to prove the multiplicative limit law:
let $(a_n)^{\infty}_{n=m}, (b_n)^{\infty}_{n=m}$ be convergent sequences of real numbers, and $X, Y$ be the real numbers $X = \lim_{n\to \...
- 174k
1
vote
2
answers
126
views
Please explain the proof of the validity of the Cauchy convergence criterion for real number sequences.
My question pertains to BBFSK, Vol I, Pages 143 and 144.
The following appears in the context of developing the real numbers as limits of sequences of rational numbers.
It is also easy to prove ...
- 3,349
1
vote
0
answers
24
views
Archimedean ordering and a greatest $r_n<x\in{M}$ shows every number less than a lower bound is also a lower bound?
This question pertains to BBFSK Vol I, page 143. The topic is the definition of the greatest lower bound of a non-empty set $M\subset{\mathbb{R}}$ which is bounded from below.
For $1<g\in\mathbb{...
- 3,349
2
votes
4
answers
57
views
Bound a natural by two consecutive powers
I'm working in the following problems: Given two naturals $m$ and $n$, there exist a natural $d$ such that
$$m^{d}\leq n \leq m^{d+1}.$$
Afterwards I need to show that: If one chooses an arbitrary $...
- 58.3k
1
vote
1
answer
257
views
well-ordering principle for natural numbers from the definition of real numbers.
Define the set of real numbers $\mathbb{R}$ by means of the following axiom: There exists a totally ordered field $(\mathbb{R},+,\cdot,\leq)$ which is Dedekind complete. We also assume that $a\leq b$ ...
- 376k
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"...
- 2,265
0
votes
2
answers
112
views
Find least positive integer $n$ which satisfies the condition
Find least positive integer $n$ which satisfies the condition
$| {0.5}^{\frac{1}{n}}-1|<10^{-3} $
$0.9^{n}<10^{-3}$
What I have tried is I have split the first inequality took log on both ...
- 26.9k
1
vote
1
answer
1k
views
Application of the Archimedean Property
Prove if that $0<a<b$ where $a,b \in \mathbb{R}$ then there exist some $n \in \mathbb{N}$ such that $\frac{1}{n} < a$ and $b < n $
The question states to use the Archimedean Property; If $...
- 83.9k
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 ...
- 13.4k