All Questions
5
questions
-1
votes
1
answer
62
views
A lemma for a new proof for the existence of a rational between two arbitrary real numbers
I had in mind to prove the theorem "There exists a rational number between each two arbitrary Real numbers" and towards the end, I happened to need to prove this: $$$$
If we have the two ...
3
votes
1
answer
663
views
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
1
vote
0
answers
80
views
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
$1.$ Every subsequence of $(s_n)$ has a further subsequence that ...
- 2,758
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
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