All Questions
Tagged with real-numbers real-analysis
1,735
questions
1
vote
1
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37
views
Shifting Index of Recursive Sequence
If I have a recursive sequence defined by:
$a_0 = 7$
$a_n = a_{n-1} + 3 + 2(n-1),$ for $n \geq 1$
How is this recursive sequence the same as the one above. Isn’t $n+1 \geq 2$?
$a_0 = 7$
$a_{n+1} = a_{...
- 129
0
votes
0
answers
49
views
Logical consistency in proof for Real Cauchy sequence implies convergence
I have doubts about this proof I reproduced from a text I have been following.
Any help in full would be appreciated as it would really help me to get more familiar with the trickiness of analysis ...
- 139
2
votes
1
answer
71
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Showing the supremum squares to 2 [duplicate]
In class, my real analysis teacher defined $\sqrt 2 = \sup \{x \in \mathbb{R} | x^2 < 2\}$ and left proving that this definition implies $\sqrt 2 ^2 = 2$ as an exercise. But I haven't been able to ...
- 127
1
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0
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31
views
Field isomorphism between copies of $\mathbb R$
$
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\tld}{\tilde}
\newcommand{\tldt}{\mathbin{\...
- 4,252
2
votes
1
answer
31
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Is the following method for proving density of irrational numbers in real numbers without using rational numbers density in real numbers rigorous?
The motivation for this question is:
I told my friend to use:
$\forall x_{1}, x_{2} \in \mathbb{R}, x_{1} < x_{2}, \exists r \in \mathbb{Q}: x_{1} < r <x_{2}.$
To prove:
$\forall x_{1}, x_{2} ...
- 1,387
3
votes
2
answers
100
views
Imagining Rational Cauchy Sequences as Dancing Around a Real Number Instead of Converging to One
I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals.
Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of ...
- 171
1
vote
1
answer
44
views
The product of a Dedekind cut and its inverse equals one
Exercise 1.11 of the textbook Real Mathematical Analysis by Pugh asks what the inverse of a cut $x=A|B$ is ($x$ being positive).
The first step is to show that $x^*=C|D$ where $C=\{1/b: b\in B \space\&...
user1327236
0
votes
0
answers
33
views
Transcendental nature of natural log for proof validity?
I am following Understanding Analysis by Stephen Abott. I read a well bit into the book but I decided to go through and do the exercises through the book because I felt as if I wasn’t being rigorous.
...
- 1
0
votes
0
answers
22
views
Majorization and second largest probability
I have a vector $p_1\geq p_2\geq\cdots p_m\geq0$ such that $\sum_{i=1}^mp_i=1$. Majorization is defined as follows: For vectors $ x, y \in \mathbb{R}^n $, $ x \prec y $ if $\sum_{i=1}^k x_{[i]} \leq \...
- 147
3
votes
1
answer
95
views
One variable inequality with parameter
Using the power-mean inequality, one can prove that for all $p\geq 1$ and for all $x\geq 0$ we have that $$(x^{p+1}+1)^2\geq \left(\frac{x^2+1}{2}\right)^p(x+1)^2.$$
The wolfram suggests that this ...
- 379
3
votes
1
answer
73
views
Whether the given function is one-one or onto or bijective?
Let $f:\mathbb{R}\to \mathbb{R}$ be such that
$$f(x)=x^3+x^2+x+\{x\}$$ where $\{x\}$ denotes the fractional part of $x$. Whether $f$ is one-one or onto or both?
For one-one, we need to show that if $f(...
- 4,500
1
vote
0
answers
53
views
On the axioms of the real number system as stated in Apostol’s textbook [duplicate]
The typical axiom system for the real numbers states that the real numbers satisfy the axioms of an algebraic field plus a few others.
In the mathematical analysis textbook by Apostol, the axioms are ...
user1327236
0
votes
1
answer
54
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Tao: Positive/negative real numbers are defined as positively/negatively bounded away Cauchy sequences. Those who start negative and turn positive -?
Quote from Tao's Analysis 1:
Definition 5.4.3 A real number $x$ is
said to be positive if and only if it can be written
as $x = \lim_{n \to \infty} a_n$ for some Cauchy sequence
$(a_n)_{n=1}^{\...
2
votes
2
answers
94
views
$| |x + y|^p - |x|^p | \leq \epsilon |x|^p + C |y|^p$
I want to demonstrate that: Let $1 < p < \infty$; for any $\epsilon > 0$, there exists $C = C(\epsilon) \geq 1$ such that for all $x, y \in \mathbb{R}$, we have
$$ | |x + y|^p - |x|^p | \leq \...
- 43
-2
votes
0
answers
49
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How does distance work? [duplicate]
I'm reading a book about PDEs and it opens with a discussion of set theory and limit points and whatnot and this got me thinking about how we define distance (since it uses neighborhoods to talk about ...
