All Questions
Tagged with real-numbers analysis
216
questions
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49
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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
1
vote
0
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31
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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
0
votes
1
answer
52
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Why is compactness necessary for IVT
The version of IVT I am working with is as follows:
Let $f\colon [a,b] \to \mathbb{R}$ be a continuous function from a compact subset of the real line, and suppose that $f(a) \lt 0$ and $f(b) \gt 0$ . ...
- 145
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29
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Proving convergence of rational function from first principles
The full question and solutions are,
here
However, the bit I am confused on is how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here ...
- 11
2
votes
2
answers
183
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Finding an irrational number between two given irrational numbers constructively
Statement:
Let $$X=\{(a,b) \in \mathbb{R} \setminus \mathbb{Q} \times \mathbb{R} \setminus \mathbb{Q}:a<b\}$$
There exists a function $f:X \rightarrow \mathbb{R} \setminus \mathbb{Q}$ such that for ...
1
vote
1
answer
59
views
Prove that there exists a $x>0$ such that $x^x = c$ WITHOUT using Intermediate Value Theorem.
I want to prove that there exists $x>0$ such that $x^x = c$ for $c>1$, and that it is unique. I have no idea how to begin, and I want to try to prove it without using the intermediate value ...
- 337
1
vote
1
answer
97
views
Validity of the proof $\mathbb R$ is uncountable
My attempt: take the interval $H=(0,1)$. Assume $(0,1)$ is countable, then there exist a bijection from $\mathbb N$ to $H$ where. Define elements of $H$ as a monotonically increasing sequence $\{a_i\}...
- 135
0
votes
0
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31
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Constructing a compact set of real numbers whose limit points form a countable set
I know this question has been answered before but I'd just like to verify that my solution is correct:
Consider a sequence $x_n = \frac{2^n - 1}{2^n}$ for $n = 0, 1, 2, \dots $ so the sequence looks ...
- 303
2
votes
1
answer
80
views
Is this following inequality with increasing powers of the components true for small $x$? And if yes, what's the positive constant $C?$
Let $0< k_1\le k_2\dots \le k_m, k_i \in \mathbb{N}, k_1 \text{ an even positive integer }. f(x_1\dots x_m):=\sum_{i=1}^{m}{x_i}^{k_i}.$ I wanted to prove, if possible, that in a small enough ball $...
- 2,354
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49
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About the Intermediate Value Theorem for Derivatives [duplicate]
I have no idea about the seemingly obvious problem below.
Problem. If a continuous function $f(x)$ is differentiable on the set of real numbers, and its derivative is always zero when $x\in \mathbb{Q}$...
- 119
0
votes
0
answers
68
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How does Dedekind axiom imply continuity axiom
I am trying to understand a theorem that proves that the supremum axiom, Dedekind axiom, and continuity axiom are all equivalent. I have trouble understanding one point in the proof that DED implies ...
- 11
3
votes
1
answer
279
views
Why can we not write the reals as a countable union of sets
I understand that the reals are not countable, what goes wrong with this? $$\mathbb{R} = \bigcup_{n=1}^\infty (-n,n) $$
- 77
2
votes
0
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146
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Can the so-called completeness of real numbers be understood as closure under limits in the real number system?
Source of background information：《The Real Analysis Lifesaver》ISBN:9780691172934
P37: “the axiom of completeness”—here, completeness is just another word for the least upper bound/greatest lower ...
- 89
2
votes
1
answer
329
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What is the measure of $A$ and $B$ which partition the reals into two subsets of positive measure?
This is a follow up to this and this post. I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.
...
- 1
3
votes
2
answers
95
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Is $\mathbb{R}\cup\{-\infty,+\infty\}$ the categorical co-completion of $\mathbb{Q}$
Seeing $\mathbb{Q}$ as an ordered set, the colimit of a diagram $D:\mathcal{I} \to \mathbb{Q}$, when it exists, is just $\operatorname{colim}D \cong \operatorname{sup}_iD(i)$.
It seems to me that ...
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