All Questions
Tagged with real-numbers analysis
12
questions
49
votes
7
answers
8k
views
Is the real number structure unique?
For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.
In my analysis class, our book ...
- 2,233
1
vote
2
answers
84
views
Proving some statements only by the definition of Real numbers.
Let $f \colon \ [0.1] \to \mathbb R$ is monotonically increasing function and $f(0)>0$ and $f(x)\neq x $ for all $x\in [0,1]$. $$A=\{x\in [0,1] : f(x)>x \}$$
We know: every non-empty subset ...
- 3,381
6
votes
2
answers
1k
views
Prove Lagrange's Identity without induction
Prove Lagrange's Identity without induction.
$$
\sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2
$$
I ...
- 13.5k
6
votes
1
answer
2k
views
approximate irrational numbers by rational numbers
I want to prove this below:
(1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
- 1,519
5
votes
1
answer
1k
views
Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma
Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
- 5,591
4
votes
2
answers
4k
views
Show $(a, b) \sim\Bbb R$ for any interval $(a, b)$ [duplicate]
Show $(a, b)$ has the same cardinality as $\Bbb R$ for any interval $(a, b)$.
From the previous chapters of the book (Understanding Analysis by Stephen Abbott), I know that $(-1,1)$ has the same ...
- 1,689
4
votes
2
answers
99
views
Solve the integral? [closed]
Help me? please How to solve this integral?
$$\int\frac{1+x^2}{1+x^4}\,dx$$
- 41
2
votes
5
answers
3k
views
Direct proof of Archimedean Property (not by contradiction)
I looked at the proof of Archimedean Property in several places and, in all of them, it is proven using the following structure (proof by contradiction), without much variation:
If $\space x \in \...
- 350
2
votes
1
answer
330
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
0
votes
3
answers
892
views
Suppose x and y are two arbitrary, distinct (unequal) real numbers. Prove that there exists a rational number q between x and y.
I thought this proof was much simpler than it actually is. I used an Axiom that states, "if $p$ and $q$ are real numbers, then there is a number between them, i.e: $$\frac{(p + q)}{2}$$
My attempt at ...
- 527
0
votes
1
answer
235
views
Why does this bijection from R2 to R not work?
Consider (x,y) in R2 with x,y both in (0,1). Write x as some decimal $x=0 .a_1a_2a_3...$ and $y=0.b_1b_2b_3b_4...$
Now, write z in R as $a_1b_1a_2b_2a_3b_3...$.
If y and x are both finite, pad the ...
- 11
0
votes
2
answers
85
views
If for all $x<y$ in $E$ there is $z\in E$ such that $x<z<y$, then $\overline{E}^\circ\neq \emptyset.$
Let $E\subset \mathbb{R}$ infinite such that for all $x<y$ in $E$ there is $z\in E$ such that $x<z<y.$ True or false? $\overline{E}$ has non-empty interior.
Attempt. I believe the answer is ...
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