All Questions
15
questions
3
votes
0
answers
112
views
Proof that for all nonzero real numbers $a$, $\frac{1}{a}$ is nonzero
I was wondering if someone could check my proof that "For all $a\in\mathbb{R}$, if $a\neq 0$ then $\frac{1}{a}\neq 0$". The definitions/assumptions I am basing the proof off of come from &...
- 31
-4
votes
2
answers
125
views
How can the reals be the set of all points on a number line when there exist non-constructible reals? [closed]
We are given the intuition that the reals form all the numbers on the numberline. However, this intuition wasn't working for me as the existence of non-constructible reals seems to me to imply that ...
- 124k
0
votes
1
answer
81
views
Characterizations of the reals
I know that one characterization of the reals is that it is the only Dedekind-complete ordered field. Are there any other characterizations of the reals as a field?
- 4,057
1
vote
0
answers
67
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What's the proof that the only Dedekind-complete field is the reals? [duplicate]
I know that the field of the rational numbers is ordered but not Dedekind-complete. What's the proof that the only Dedekind-complete field is the reals?
- 4,057
0
votes
1
answer
756
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Prove that real multiplication distributes over addition
The distributive property of real numbers states that $“$for all $a, b, c \in \mathbb{R}$, we've $a⋅(b + c) = a⋅b + a⋅c$ and $(b + c)⋅a = b⋅a + c⋅a”$. How to prove this field property of real numbers? ...
CommunityBot
- 1
0
votes
2
answers
45
views
Proving a set X is dense in [0,1] equivalence relation [duplicate]
Let the relation in $\mathbb{R}: x \equiv y \ \mbox{mod} \ \mathbb{Z}$, when $x-y \in \mathbb{Z}$. For each $n \in \mathbb{N}$, let $x_n \in [0,1)$ such that $x_n \equiv \sqrt{n} \ \mbox{mod} \ \...
- 58.3k
6
votes
2
answers
118
views
Two "different" definitions of $\sqrt{2}$
In Walter Rudin's Principles of Mathematical Analysis (3rd edition) (page 10), it is proved that
for every $x>0$ and every integer $n>0$ there is one and only one positive real $y$ such that ...
CommunityBot
- 1
-2
votes
1
answer
48
views
Construct a sequence in A that converges to the supremum of A [closed]
It is similar to this question that I learned quite a bit from:
Showing the set with a $\sup$ has a convergent sequence
But I want to ask how can I construct an example of (Sn).
i.e.
If A is a ...
- 5,490
0
votes
2
answers
71
views
Proving $x^{r} \cdot x^{s} = x^{r + s}$ provided two facts
I have the following two facts:
For positive numbers $a$ and $b$ and natural numbers $n$ and $m$, we have $a = b$ if and only if $a^{n} = b^{n}$ if and only if $a^{1/m} = b^{1/m}$.
For a ...
CommunityBot
- 1
6
votes
2
answers
392
views
Path From Positive Dedekind Cuts to Reals?
Don't spend a lot of time on this. I'm certain I could bang it out myself; but maybe there's an answer out there that someone already knows.
Say we use Dedekind cuts to construct the reals. Addition ...
- 11.5k
1
vote
2
answers
269
views
real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ . [duplicate]
Problem :
Verify if the statement if true of false -
The set $S$ of all real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ .
I think this ...
- 1,254
2
votes
1
answer
449
views
Question on the archimedean property
Let $a,b \in \Bbb ℝ$. Suppose that $a>0$. Prove that there is some $n\in \Bbb N$ such that $b\in[-na, na]$.
I understand how the Archimedean Property can be used to prove this statement if $b$ is ...
- 358
1
vote
3
answers
2k
views
Arithmetic Operations with Infinities in Real Analysis
Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers
This is the usual reason given when asked why we can't perform the usual arithmetic ...
- 43.8k
3
votes
1
answer
180
views
Dedekind's Cuts Lemma
I'm studying Dedekind's Cuts and his construction of Real numbers from the Rational ones. Here we are allowed to use $\Bbb{Q}$ as an ordered field and all all its properties (Archimedean Property, his ...
- 1,533
0
votes
1
answer
70
views
Related Zorn's lemma proof?
Let $S$ be a partially ordered set, with the additional property that every chain $s_0\le s_1 \le s_2 \le...$ has an upper bound in $S$ (i.e. there is some $t$ in $S$ such that $s_n \le t$ for all $n$)...
- 65.3k