All Questions
Tagged with real-numbers proof-writing
127
questions
-1
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1
answer
142
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How to prove given sequence converges to 0? [duplicate]
I am given a sequence {$a_{n}$} of nonzero numbers converges to infinity. How can I use this to prove that the sequence {$\frac{1}{a_{n}}$} converges to 0?
I can intuitively see why $\frac{1}{\infty}$...
- 79
1
vote
2
answers
349
views
Explain this "contradiction" of the proof of $x > 0$ iff $x \in \mathbb{R}^{+}$
I am working through Apostol's Calculus I and just read about the order axioms. He presents a new undefined concept called positiveness, gives the axioms and then defines symbols $<, >, \leq, \...
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 ...
0
votes
2
answers
277
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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
0
votes
2
answers
92
views
Validity of Proof for 'Possibility of Subtraction' from Apostol 1
I attempted a proof before reading the solution Apostol provides. I don't think it is valid but I am trying to determine why
Theorem 2, Possibility of Subtraction: Given $a \text{ and } b$, there is ...
3
votes
2
answers
69
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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
2
votes
4
answers
110
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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 \...
user548054
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.
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1
answer
388
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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 ...
- 3,569
0
votes
0
answers
56
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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\...
- 25
1
vote
2
answers
660
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 ...
- 2,193
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$ ...
- 215
3
votes
1
answer
996
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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 ...
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votes
2
answers
49
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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 ...
- 4,665
2
votes
2
answers
3k
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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
1
vote
0
answers
1k
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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 ...
- 3,306
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{_{...
- 414
17
votes
2
answers
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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
1
vote
5
answers
2k
views
Proof that all real numbers have a rational Cauchy sequence?
I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known ...
- 1,681
0
votes
1
answer
138
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Proving $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$
I am trying to show $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$ using the Cantor Bernstein method. Here is my proof so far:
Let $f: \mathbb{R}\to\mathbb{R}^\mathbb{N}$ be defined as for each $r_n\in\...
- 1,088
-1
votes
2
answers
145
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prove quadratic polynomial has no real roots
The problem asks me to prove that a polynomial $f(x)=x^2+ax+b$ has no real roots for some $a,b \in \Bbb{R}$
I started by assuming that $f(x)=x^2+ax+b$ has real roots and therefore the determinant $a^...
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3
votes
1
answer
666
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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
2
votes
3
answers
353
views
A positive real number $x$ with the property $x^3=3$ is irrational.
I have the following problems:
1) There exists a positive real number $x$ such that $x^3=3$.
2) A positive real number $x$ with the property $x^3=3$ is irrational.
My Idea for 1) would be (there ...
- 97
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
3
votes
4
answers
944
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
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3
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3k
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Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
(1) I start with an obvious fact:$$0 = 0$$
(2) Add $(-1)$ to both sides of the equation:
$$0 + (-1) = 0+ (-1)$$
(3) Zero is the ...
- 3,719
1
vote
1
answer
199
views
Proving equinumerosity of two sets
My question reads:
Prove that if $a<b$ are real numbers, then (i) $(a,b)$~ $(0,1)$ and (ii) $[a,b]$~ $(0,1)$
Now, we have proven that $(0,1)$~ $\mathbb{R}$, so should we consider this we finding ...
- 347
0
votes
1
answer
43
views
Equivalence of Cauchy sequences implies corresponding inequality
Let $X:=\{(q_n)_n \subset \mathbb Q: (q_n)_n \mbox{ is Cauchy} \}$. Then, for $(q_n)_n, (r_n)_n\in X$, $(q_n)_n \sim (r_n)_n$ means that $(q_n)_n$ and $(r_n)_n$ are equivalent if for every $0<\...
- 9,708
-3
votes
1
answer
67
views
Proving: $a^2 < b^2 ⇔ |a| < |b|$
I started studying mechanical engineering and it works perfectly fine for me but i stumbled across this problem:
$$a^2 < b^2 ⇔ |a| < |b|$$
I found a solution but that took me a full piece of ...
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-1
votes
1
answer
82
views
Polynomial proof in Real numbers [closed]
how to prove that there exists a 2 variable polynomial which is bounded below and the range of values is strict subset from the $\mathbb{R}$.
- 3
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
0
votes
2
answers
309
views
Conceptual question about trichotomy of real numbers
I was thinking that it is possible to answer this Math.SE question here using the trichotomy of the real numbers but I got some logic trouble.
The question was. Prove that for all $x,y > 0$ ...
- 2,512
0
votes
1
answer
67
views
For any $\varepsilon >0$, there is an $n\in\mathbb{N}$ such that $\dfrac {1} {n} < \varepsilon$.
Archimedean Property. For any $x,\varepsilon\in\mathbb{R}$, $\varepsilon>0$, there is an $n\in\mathbb{N}$ such that $n\varepsilon>x$.
Corallary. For any $\varepsilon >0$, there is an $n\in\...
user295645
1
vote
3
answers
138
views
How can I prove that the first three decimal digits of a real number between 0 and 1 can be equal to the reciprocal of 2 to the power of that number?
This is the challenge problem at the end of Chapter 1 of Solow's How to Read and Do Proofs.
The problem:
Find a counter-example to the following statement: “If x is a
positive real number ...
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0
votes
6
answers
131
views
Prove $5\sin x - 4\cos x < 9$
My attempt at proving is as follows:
If
$-1 \leq \sin x \leq 1$ and $1 \leq \cos x \leq 1$
then
$5(-1) - 4(-1) = -5 + 4 = -1$
$5(1) - 4(1) = 5 - 4 = 1$
so
$-1 \leq 5 \sin x - 4\cos x \leq 1 &...
- 31
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 ...
- 2,756
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
0
votes
2
answers
62
views
Struggling of a proof of cardinality
I'm working on a homework problem and I'm not sure I'm understanding how to construct the proof. I have done work on it, but it's not coming together. The problem, "Let $A \subseteq (0, 1)$ be a set ...
- 599
0
votes
3
answers
62
views
Can someone point out the holes in my proof using well ordering, archimedean and completeness to prove for a given set E, inf(E) = -sup(E)
Let, $$E\subset \mathbb R$$ be a non empty bounded above set. Define $$ -E = \{ -x : x \in E \}.$$ Then, $\operatorname{inf} (-E) = -\operatorname{sup}(E)$.
Proof - It follows from the completeness ...
- 319
0
votes
1
answer
67
views
Trouble proving the floor function is onto with the domain being all real numbers
I need to prove that for the mapping $f : \mathbb{R} \mapsto \mathbb{Z} $ given by $ f(x) = \lfloor x \rfloor$, $f$ is onto. I know how I would do it if both the domain and codomain were both $\mathbb{...
0
votes
2
answers
1k
views
How to show/proof that the union of two non empty subsets of ${\Bbb R_{}}$ has a least upper bound?
We have two sets ${E}$ and ${T}$, that are non empty subsets of ${\Bbb R_{}}$ and are bounded above.
How can I prove that,
${E}$ ${\cup}$ ${T}$ has a least upper bound (supremum), and that ${\sup(E\...
- 414
0
votes
2
answers
2k
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How can I proof the infimum and supremum of this set?
$E = \{{x+y : x,y \in\Bbb R_{>0}}$}
I was able to figure out that this set does not have a supremum, but I am not able to prove it. Also, how can I prove the infimum of this set ?
This is my ...
- 414
0
votes
1
answer
896
views
How to prove comparability property & writing its proof
I am given the relation in $\mathbb{R}$: $xRy$ if $x\le 2^y$. I want to prove this has the comparability property, so I know I start with let $x,y\in \mathbb{R}$. Then I need to show either $xRy$ or $...
- 1,088
0
votes
2
answers
384
views
Proving some properties of real numbers using predicate logic
I am trying to understand how some basic properties of the real numbers can be proved from axioms expressed in predicate logic.
I start by accepting the field axioms of real numbers, in addition to ...
- 777
1
vote
2
answers
1k
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Help determining if a finite subset of $\mathbb R$ is closed and bounded.
If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
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3
votes
2
answers
3k
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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 ...
- 7,297
0
votes
1
answer
407
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Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.
As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations.
...
- 1
0
votes
0
answers
20
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Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
I am trying to formally prove:
$ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
where n is an integer, and x and y are natural numbers.
It is obvious that, when $\frac xy$ is ...
- 101
7
votes
5
answers
2k
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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