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
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
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
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
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
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
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
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
2
votes
4
answers
743
views
How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?
Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$.
One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
- 1,893
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
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
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
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)...
2
votes
1
answer
73
views
Using the *compact* property of subsets in $R$ to prove Bolzano-Weierstrass Theorem
I have been asked to prove the Bolzano-Weierstrass Theorem with respect to a bounded sequence of real numbers by using the fact that closed and bounded subsets of $R$ are compact.
There is a hint ...
- 346