All Questions
Tagged with real-numbers proof-writing
127
questions
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 ...
- 80.2k
2
votes
4
answers
110
views
Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$?
Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$?
And is it possible to do it without using calculus?
$$2^x=x+1.$$
Here is my attempts:
$2^x>0 \...
- 47k
0
votes
3
answers
103
views
Given $x^2=2$ prove for any rational number $\frac{p}{q} < x$,there exists $\frac{m}{n}$ such that $\frac{p}{q}<\frac{m}{n}<x$
Without using limits or the definition of irrational numbers, how do you solve this? I was thinking proof by contradiction, but I keep running into problems.
- 1,287
0
votes
1
answer
388
views
Problem in proof of -"A net has $y$ as a cluster point iff it has a subnet which converges to $y$"
Directed Set:
We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$
such that
(i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$;
(ii) x ≤ x for each x ∈ $\omega$;
(iii) for ...
CommunityBot
- 1
0
votes
0
answers
56
views
Proving closed interval [0,1] has same cardinality as Real Numbers [duplicate]
I want to prove that the closed interval [0, 1] has same cardinality as Real Numbers. I was able to figure out (0, 1), but need help with proving the closed part. What I have so far is:
the function $...
- 819
0
votes
2
answers
2k
views
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:
$$
\forall x\in\...
- 126k
1
vote
4
answers
8k
views
Prove $(-x)y=-(xy)$ using axioms of real numbers
Working on proof writing, and I need to prove
$$(-x)y=-(xy)$$
using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...
- 844
1
vote
2
answers
659
views
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 ...
- 176
1
vote
1
answer
2k
views
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$ ...
CommunityBot
- 1
3
votes
1
answer
995
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 ...
- 2,453
0
votes
2
answers
49
views
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 ...
- 6,653
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 +$\...
- 737
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|\...
- 334k
1
vote
0
answers
1k
views
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 ...
- 176k
3
votes
2
answers
807
views
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{_{...
- 688