All Questions
36
questions
6
votes
3
answers
789
views
Where is the mistake in the argument in favor of the (erroneous) claim "every Dedekind cut is a rational cut"?
A cut is a set $C$ such that:
(a) $C\subseteq \mathbb Q $
(b) $C \neq \emptyset $
(c) $C \neq \mathbb {Q} $
(d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $
(e) for all $c\...
- 1,796
2
votes
1
answer
97
views
How do I write this Theorem with quantifiers?
Here is the theorem from Steven Abbot's Understanding Analysis.
Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \...
- 149
3
votes
1
answer
90
views
Proposition 5.4.9. Analysis I - Terence Tao.
Proposition 5.4.9 (The non-negative reals are closed). Let $a_1, a_2, a_3, \ldots $ be a Cauchy sequence of non-negative rational numbers. Then $\text{LIM}_{n \to \infty}a_n$ is a non-negative real ...
- 1,454
4
votes
1
answer
273
views
The most explicit way of partitioning the reals into two dense subsets with positive measure
In @UmbertoP's response to the question, "Partition of real numbers into dense subsets of positive measure," the answer is understandable to a advanced undergraduate; however, I have ...
- 1
0
votes
0
answers
45
views
Is given statement indeed true?
I saw saw the statement I am trying to prove in "a proof of lub property" in a Mathematics Stack Exchange post. The statement I am trying to prove is :
$∃\ b ∈ \mathbb {R},(b < a\
\text { ...
- 556
0
votes
0
answers
29
views
Confused on where it says that the inverse is unique on the Property of Existence of a multiplicative inverse [duplicate]
For every number $ a \ne 0 $, there is a number $ a ^{-1} $ such that
$$ a . a^{-1} = a^{-1} . a = 1. $$
This line is from Calculus by Michael Spivak.
I wanted to prove $ (ab)^{-1} = a^{-1}b^{-1} $ ...
0
votes
0
answers
53
views
Can and can’ts of proving inequalities
I was doing the following question and wonder if my proof below is valid
Prove that $\frac{x+z}{y+z} > \frac{x}{y} \to x<y$ if and only if $x<y$
To prove that $\frac{x+z}{y+z} > \frac{x}{...
0
votes
0
answers
31
views
Suppose $(a_n)$ is a sequence of pos. real numbers such that $\lim_{n \to \infty}\frac{a_{n+1}}{a_n} = 0$. Prove that $\lim_{n \to \infty}{a_n} = 0$. [duplicate]
Suppose $(a_n)$ is a sequence of positive real numbers such that
$\lim_{n \to \infty}\frac{a_{n+1}}{a_n} = 0$
Prove that $\lim_{n \to \infty}{a_n} = 0$. (Warning: Be careful not to assume that
$(a_n)$ ...
1
vote
2
answers
93
views
Proving an expression for the limit of a certain sequence [closed]
If $(a_{n})_{n=1}^{\infty}$ is a sequence converging to $L$, with $a_n \geq 0$ for all $n$, how can I prove that $L \geq 0$ and that $(\sqrt{a_n})_{n=1}^{\infty}$ converges to $\sqrt{L}$.
I was under ...
1
vote
1
answer
74
views
Confusion about q-ary system in Zorich's Mathematical analysis
For $p \in \Bbb Z $ there is a q-ary system described which assigns to each real number x in the base q a sequence of {$\alpha_n$} such that $\sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i} \le x \lt q^{p-n} +...
- 41
2
votes
1
answer
330
views
Rudin Principles of Mathematical Analysis Theorem 6.10
Rudin is using the Riemann-Steljis integral and its assumed $\alpha$ is monotonic. Throughout this chapter Rudin has written $M_{i} = \sup f(x)$ where $x_{i-1} \leq x \leq x_{i}$ and $m_{i} = \inf f(x)...
1
vote
1
answer
86
views
Is this lemma correct?
The lemma is stated as an alternative definition for a least upper bound
$$\mathrm{sup} A - \epsilon < a$$
Let $A$ be a set where $A \subseteq \mathbb{R}$ and $\epsilon > 0$, such that for every ...
user956717
0
votes
1
answer
78
views
Clarification for the proof of uncountability of real numbers
The following is from the Understanding Analysis 2nd ed., Stephen Abbot, page 28
If we let $x_1 = f(1), x_2 = f(2)$ and so on, then our assumption that $f$ is onto means that we can write $\mathbb{R} ...
- 1,030
1
vote
1
answer
372
views
Showing that a rational number exists between any two reals
The following is from Real Analysis by N. L. Carothers.
Theorem: If $a$ and $b$ are real numbers with $a < b$, then there is a rational number $r$ $\in \mathbb{Q}$ with $a < r < b$.
Proof: ...
- 1,030
1
vote
0
answers
117
views
Guidance on a proof regarding the uniqueness/existence of the nth root of x.
So in a previous proof I had to show that given $y\in\mathbb{R},n\in\mathbb{N}$ and $\epsilon >0$, show that for some $\delta>0$, if $u\in\mathbb{R}$ and $|u-y|<\delta$ then $|u^n-y^n|<\...
- 3,431
1
vote
1
answer
153
views
l don't understand the way $\epsilon$ and $\delta$ are being used in this question about proving $|u-y|<\delta\Rightarrow|u^n-y^n|<\epsilon$
The full statement is: Given $y\in\mathbb{R},n\in\mathbb{N}$ and $\epsilon>0$, show that for some $\delta>0$, if $u\in\mathbb{R}$ and $|u-y|<\delta$ then $|u^n-y^n|<\epsilon$.
Ultimately, ...
- 3,431
2
votes
2
answers
127
views
How should I interpret this diagram showing the bijection from $(a,b)$ to $\mathbb{R}$
In Chapter 1 of Pugh's Real Mathematical Analysis, Pugh gives the following picture:
I'm aware of other proofs to this like this one: bijection from (a,b) to R
but I'm interested in understanding how ...
- 363
2
votes
2
answers
230
views
Explanation of Shakarchi's proof of 1.3.4 in Lang's Undergraduate Analysis
I'm currently working through Lang's Undergraduate Analysis, and trying to understand Rami Shakarchi's proof of the following:
Let $a$ be a positive integer such that $\sqrt a$ is irrational. Let $\...
- 260
1
vote
2
answers
68
views
Help with a proof of a consequence from the axioms of addition and multiplication
While reading through Analysis 1 by Vladimir A. Zorich, I encountered this proof which has this 1 step I can't understand. Here is the consequence and the proof:
For every $x\in \mathbb R$ the ...
- 537
2
votes
2
answers
148
views
Are real numbers enough to solve simpler exponential equations such as $2^x=5$, $(1/e)^x=3$, and $\pi^x=e$?
How to prove that solutions of simpler exponential equations (*) are real numbers?
In other words, how to prove that set of real numbers is enough to solve something like $2^x = 5,$ or $(\frac{1}{e})^...
- 1,276
0
votes
1
answer
81
views
A question about the expansion in base $g$ of real numbers
I'm reading the proof of Theorem 7.11 from textbook Analysis I by Amann/Escher.
For $x \in [0,1)$ the authors define recursively the sequence $(x_k)_{k \in \mathbb N}$ as follows:
In the ...
- 17.6k
1
vote
4
answers
136
views
How do I prove that for continuous $f$, if $\forall x \in \mathbb{Q}, \, f(x) = f(1)^x$ then $\forall x \in \mathbb{R}, \, f(x) = f(1)^x$?
Given the differentiable/continuous real-valued function $f(x)f(y) = f(x + y)$ I got so far as to show that $\forall x \in \mathbb{Q}, \, f(x) = f(1)^x$.
I am trying to show that because $f$ is ...
- 169
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
3
votes
5
answers
989
views
Proof explanation of $``\exists x\in\mathbb{R}$ with $x^2=2"$
Can someone please help me break down the proof below from $(*)$ onwards. I'm lost at what is going on and where the proceeding steps are coming from. Is this a proof by contradiction? Why are we ...
user503154
0
votes
2
answers
131
views
Showing $\sup A\ge 2$
I am struggling to understand the proof in the textbook.
Let $A$ contain elements $x$, and $x$ is real number which satisfies $x^2 < 2$. Let $\sup A = r$, and show that $r^2 \ge 2$.
In the ...
- 3,659
3
votes
4
answers
90
views
Let $a^2<2, b=2(a+1)/(a+2)$. Show $b^2<2$ (assignment)
It is a part of my assignment.
$$ \text {Let }a^2<2, \quad b=2\frac {(a+1)}{(a+2)}\quad \text{ Show } b^2<2$$
I already proved that a
But, I am struggling to prove $b^2<2$.
My lecturer ...
- 3,659
3
votes
2
answers
807
views
Proving that a sequence converges to L
Given a sequence $(a_{n})_{n=1}^{\infty}$ that is bounded. Let $L \in R$. Suppose that for every subsequence $(a_{n{_{k}}})_{k=1}^{\infty}$ , either $$\lim_{k \to \infty}a_{n{_{k}}} = L$$
or $(a_{n{_{...
- 414
2
votes
2
answers
1k
views
$r = \frac{m}{n} = \frac{p}{q} \Rightarrow (b^m)^{1/n} = (b^p)^{1/q}$; help with proof
This is from Baby Rudin chapter 1, exercise 6, and I'm using the unofficial answer key found here: https://minds.wisconsin.edu/handle/1793/67009
There is also this proof here: Prove that $(b^m)^{1/n} ...
- 1,001
1
vote
1
answer
174
views
Does this mean the preimage of vertical lines or the preimage of horizontal lines?
Consider the following statement:
Prove that it is possible to write $\Bbb R$ as a union $\Bbb R= \bigcup_{i\in I} A_{i}$ where $A_{i} \cap A_{j}= \emptyset$ if $i\neq j$, $i,j \in I$,and such that ...
- 4,223
3
votes
1
answer
602
views
Proof explanation for the statement that $\Bbb R$ can be partitioned into a union of uncountable sets where the index set is also uncountable
Consider the following statement:
Prove that it is possible to write $\Bbb R$ as a union $\Bbb R= \bigcup_{i\in I} A_{i}$ where $A_{i} \cap A_{j}= \emptyset$ if $i\neq j$, $i,j \in I$,and such that ...
- 4,223