All Questions
Tagged with real-numbers proof-writing
15
questions
17
votes
2
answers
19k
views
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
7
votes
5
answers
2k
views
Is this direct proof of an inequality wrong?
My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
- 81
5
votes
1
answer
6k
views
For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?
If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$?
Please help me to clarify the above. Thanks in advance.
- 1,975
3
votes
2
answers
1k
views
$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon>0$. How to prove $x \le y$? [closed]
$x,y \in \mathbb R$ are such that $x \lt y + \epsilon$ for any $\epsilon \gt 0$ Then prove $x \le y$.
2
votes
1
answer
86
views
Alternative proof of $a\times0= 0$
I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
2
votes
2
answers
3k
views
If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
(a) If sup A < sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.
I have argued that if sup A $\lt$ sup B, then choose an $\epsilon>0$ such that sup A +$\...
- 3,286
0
votes
2
answers
277
views
Real Analysis Inequality Proof Involving Reals and Rationals $0 < |r - q| < \varepsilon$
I'm having difficulties making progress in proving:
$$\forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$
To clarify, $r$ is a real number and $q$ is a ...
- 79
6
votes
4
answers
11k
views
How to prove that every real number is the limit of a convergent sequence of rational numbers?
Here is my procedure:
so we want to prove $\forall r\in \mathbb{R},$ there exists a sequence $q_n$ of rationals such that $\forall\epsilon\gt 0,$ there exists a $N$ such that $n\gt N\implies |q_n-r|\...
- 2,758
4
votes
2
answers
172
views
Proving the density of a function in an interval.
I am reading Steven G. Krantz's Real Analysis and Foundations and came across this problem.
Problem: Let $\lambda$ be a positive irrational real number. If $n$ is a positive integer, choose by the ...
- 290
3
votes
1
answer
993
views
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
3
votes
0
answers
190
views
My first simple direct proof (very simple theorem on real numbers). Please mark/grade.
What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Let $I = [a,b]$ be a non-empty closed ...
- 1,321
2
votes
1
answer
214
views
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 ...
- 347
0
votes
1
answer
364
views
Proof of Dedekind cuts.
This is my definition for Dedekind cuts:
A subset α of Q is said to be a cut if:
$α$ is not empty,$α\neq \mathbb{Q}$
If $p \in α,q\in\mathbb{Q}$,and $q<p$,then $q\inα$.
If $p\in α$,then $p<r$ ...
- 97
0
votes
4
answers
173
views
Prove that, there are 4 real roots of system of equations: $\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$
How can I prove that, there are 4 real roots of this system of equation?
Solve for real numbers:
$$\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$$
My attempts:
$$(7-x^2)^2+x=11 \Longrightarrow x^4 - ...
user548054
-3
votes
1
answer
67
views
Proving: $a^2 < b^2 ⇔ |a| < |b|$
I started studying mechanical engineering and it works perfectly fine for me but i stumbled across this problem:
$$a^2 < b^2 ⇔ |a| < |b|$$
I found a solution but that took me a full piece of ...
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