All Questions
Tagged with real-numbers analysis
217
questions
2
votes
2
answers
1k
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Inequality for all real numbers
I am trying to prove a hw problem from Taos Analysis 1 book. I would like some help proving the following statements if they are true which I do not necessarily believe.
Let $x,y \in \Bbb R$. Show ...
- 556
2
votes
0
answers
148
views
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
331
views
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
2
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0
answers
52
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Using Dedekind Cut to construct multiplicative inverse in $\Bbb R$
I was trying to construct a multiplicative inverse in terms of Dedekind cuts: Fix some $\alpha\in\Bbb R^*$ (that means the positive real numbers), we define the identity element $1^*$ to be $$1^*:=\{q\...
- 160
2
votes
1
answer
743
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Why is the open Set $(0,1)$ equivalent to the closed set $[0,1]$?
I understand the proof based on ordering $[0,1]$ into a set $A$ of distinct points that include $0$ and $1$ and then showing the one-to-one equivalence to $(0,1)$, but what I can't get my head around ...
- 163
2
votes
0
answers
51
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Looking for a not measurable function
I'm looking for a counter example as simple as possible.
I've learned that if X is a stochastic variable, it holds that
\begin{align}E[X | X] = X E[1 | X]\qquad (1) \end{align}
since X is measurable ...
- 394
1
vote
2
answers
215
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Example of a discrete bounded subset of $\mathbb{R}$ such that for every two points of that set there is another point between them of that set
Can anyone give an example of a subset say $A$ of $\mathbb{R}$ such that every point of that set is isolated i.e. for every $x \in A$ there exist a neighborhood say $N(x)$ containing x such that $N(x)...
- 632
1
vote
4
answers
84
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Let $A \subseteq \mathbb{R}$. Prove or Disprove that A disconnected implies that $\overline{A}$ is also disconnected.
Having trouble on proving this, I was thinking of doing the contrapositive.
Suppose $\overline{A}$ is connected i.e. $(\forall B, C \subseteq \overline{A}) (B \cup C \neq \overline{A})$ or $(\overline{...
- 629
1
vote
4
answers
63
views
Let $f$ be a differentiable function in $a$, which belongs to the real ones, find the limits
Let $f$ be a differentiable function in $a \in \Bbb R$ , find the limits
$$\lim_{x\to a} \left( \frac{f(x)e^x-f(a)}{f(x)cos(x)-f(a)} \right)$$ $a=0, f'(0) \ne 0$
and
$$\lim_{x\to \infty} \left( n\...
- 83
1
vote
1
answer
119
views
Finding rational points at rational distance in the plane
Take any point $p$ in the real plane.
Does there always exist a rational point at a rational distance from $p$?
(A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)
- 11.6k
1
vote
1
answer
99
views
How to prove $\langle x^{2n+1}: n\in \mathbb{N}\rangle$ is dense in $\{ f\in C([0,1]): f(0)=0\}$
I'm trying to prove that $\langle x^{2n+1}: n\in \mathbb{N}_0\rangle$ is dense in $\{ f\in C([0,1]): f(0)=0\}$ without the use of the Müntz–Szász theorem.
I know how to prove this for even exponents ...
1
vote
1
answer
1k
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Let $h:R\to R$ be continuous on$ $R satisfying $h(m/2^{n}) =0$ for all $ m \in Z $, $ n \in N $. Show that $h(x)=0$ for all $ x \in R $.
Let $h:R\to R$ be continuous on $R$ satisfying $h(m/2^{n}) =0$ for all $ m \in Z $, $ n \in N $ .Show that $h(x)=0$ for all $ x \in R $.
I think sequential criterion of continuity .but how to show $(...
- 343
1
vote
3
answers
663
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Intuition between Dedekind cut construction of real numbers
I understand the construction of real numbers via Dedekind cuts I think, the set of Dedekind cuts over $Q$ are real numbers. However, I don't see how it gives us any more understand or power than the ...
- 33
1
vote
1
answer
134
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Does a real number with this decimal expansion for $r$ and $r^2$ exist?
Does there exist a real number $0< x <1$, such that the decimal expansions of $x$ and $x^2$
are the same, starting from the
millionth term, and neither expansion has an infinite ...
- 265
1
vote
2
answers
102
views
Why the following sequence of function does not converge uniformly at $[0, \infty)$
Why the following sequence of function does not converge uniformly at $[0, \infty)$ but converge uniformly for some $a>0, [a,\infty)$
$$f_n(x) := n^2x^2e^{-nx}$$
So I know the limit function $f$ ...
- 2,157