All Questions
6
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
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 +$\...
- 3,306
2
votes
1
answer
87
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 = ...
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|\...
- 2,756
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 ...
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