All Questions
Tagged with real-numbers proof-writing
127
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Absolute Value Proof By Cases
I'm currently working through D. Velleman's How to Prove it. I have a question regarding an absolute value proof by cases (#10; section 3.5).
The question asked is to prove that:
$$
\forall x\in\...
- 25
1
vote
2
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659
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Can every real number be uniquely represented as a sum of a rational number and an irrational number $\in [0, 1)$?
I've needed to prove the transitivity of the following relation on the set of all real numbers:
$x − y$ is a rational number.
Immediately I've thought "Every real number can be uniquely ...
- 2,193
2
votes
1
answer
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Prove there is an integer larger than a given real number
A homework problem guides me to prove that there is a rational number $\frac{m}{n}$ between every two real numbers $x$ and $y$. The first step requires me to prove that there exists an integer $n$ ...
- 225
3
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1
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993
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How to prove $\sqrt{2}\in \Bbb{R}$ with Dedekind cuts?
Problem statement: Prove that $\sqrt{2}\in\Bbb{R}$ by showing $x\cdot x=2$ where $x=A\vert B$ is the cut in $\Bbb{Q}$ with $A=\{r\in\Bbb{Q}\quad : \quad r\leq 0\quad \lor \quad r^2\lt 2\}$. Denote the ...
- 1,768
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LUB (if it exists) of a complete set belongs to that set: Validity
By LUB I mean the least upper bound of the set.
And the definition of complete set I am using is that every Cauchy sequence in that set must converge in that set.
So by these two assumptions.
I cannot ...
- 4,645
2
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2
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3k
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If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
(a) If sup A < sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.
I have argued that if sup A $\lt$ sup B, then choose an $\epsilon>0$ such that sup A +$\...
- 3,286
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Let $a < b $ be real numbers and consider set T = $\mathbb Q \ \cap \ [a,b].$ Show $\sup \ T =b$
Let $a < b $ be real numbers and consider set T = $\mathbb Q\cap [a,b].$ Show $\sup T =b$
I needed help checking if my proof is correct. If it isn't correct can you please provide the correct ...
- 3,286
3
votes
2
answers
805
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Proving that a sequence converges to L
Given a sequence $(a_{n})_{n=1}^{\infty}$ that is bounded. Let $L \in R$. Suppose that for every subsequence $(a_{n{_{k}}})_{k=1}^{\infty}$ , either $$\lim_{k \to \infty}a_{n{_{k}}} = L$$
or $(a_{n{_{...
- 414
17
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2
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Proof there is a rational between any two reals
This is a problem from Rudin, but I wanted to add my own intuition to it. It uses Rudin's definition of Archimedean property. I'd just like to know if my version holds
If $x \in \mathbb R$, $y\in \...
- 1,755
1
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5
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Proof that all real numbers have a rational Cauchy sequence?
I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known ...
- 1,681
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1
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136
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Proving $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$
I am trying to show $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$ using the Cantor Bernstein method. Here is my proof so far:
Let $f: \mathbb{R}\to\mathbb{R}^\mathbb{N}$ be defined as for each $r_n\in\...
- 1,088
-1
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2
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145
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prove quadratic polynomial has no real roots
The problem asks me to prove that a polynomial $f(x)=x^2+ax+b$ has no real roots for some $a,b \in \Bbb{R}$
I started by assuming that $f(x)=x^2+ax+b$ has real roots and therefore the determinant $a^...
- 689
3
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1
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663
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Prove: If $x$ has the property that $0\leq x<h$ for every$ h>0$, then$ x=0$.
I'm going through Apostol's Calculus I introduction, and I'm trying to prove this, but I'm having a little trouble doing it. It's proposed as an exercise in section I 3.5: order axioms.
So, what I ...
- 1,155
2
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3
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353
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A positive real number $x$ with the property $x^3=3$ is irrational.
I have the following problems:
1) There exists a positive real number $x$ such that $x^3=3$.
2) A positive real number $x$ with the property $x^3=3$ is irrational.
My Idea for 1) would be (there ...
- 97
2
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1
answer
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Proving that $ ^\mathbb{Q}\mathbb{R}$~$\mathbb{R}$ using Cantor-Bernstein [duplicate]
I am trying to prove that $ ^\mathbb{Q}\mathbb{R}$~$\mathbb{R}$ .
I want to use Cantor Schroder Bernstein Theorem rather than coming up with a bijection. Any suggestions on how to get started with ...
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