All Questions
Tagged with real-numbers proof-writing
127
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
7
votes
2
answers
277
views
Baby Rudin Theorem 1.19 Step 5
I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's ...
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
6
votes
1
answer
202
views
Different definitions of the archimedean property
In some textbooks I have seen the archimedean property defined as:
for some positive real $x$, real number $y$, there exists a natural $n$ such that $nx>y$.
In other textbooks the archimedean ...
6
votes
1
answer
3k
views
Proving that the absolute value of x is greater then or equal to $0$
My Question reads:
Prove for all $x\in\mathbb{R}$, $|x|\geq\ 0$.
This is for a set theory class where we know that $\mathbb{R}$ is the set of Dedekind cuts.
For each $x\in\mathbb{R}$, we define
$|...
- 1,088
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
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
4
votes
2
answers
12k
views
Proof clarification - If $ab = 0$ then $a = 0$ or $b =0$
I came across a proof for the following theorem in Apostol Calculus 1. My question is regarding (1) in the proof, why is this part necessary? I don't see why you can't begin with (2)
Theorem 1.11
If ...
3
votes
4
answers
941
views
Why does a/b have to be in simplest form in the proof of irrationality for sqrt2
The proof of the irrationality of $\sqrt{2}$ starts with the supposition that $\sqrt{2} = \frac ab$ where $a$ and $b$ are integers. I understand that, but why is it important that $\frac ab$ is ...
- 31
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 $x \le y$.
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
3
votes
2
answers
3k
views
How to prove the power set of the rationals is uncountable?
Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
- 7,202
3
votes
3
answers
467
views
Formal proof of: $x>y$ and $b>0$ implies $bx>by$?
Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$.
What is a formal (low-level) proof of this result? Or is this property taken as axiomatic?
The motivation for this ...
3
votes
2
answers
69
views
Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.
I came across the implication
$$x < y \Rightarrow x^n < y^n$$
$$x,y>0, n\in Z^+$$
in a textbook and came up with the following proof.
Proof
Since $x<y$ the following chain of inequalities ...
- 33