All Questions
36
questions
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