All Questions
Tagged with real-numbers proof-writing
127
questions
0
votes
1
answer
22
views
How to more formally prove this inequality
This is a simple problem I came up with while doing another problem:
Given: $n < (n + \frac{1}{2}) < y < (n + 1)$
Prove: $|y - n| > |y - (n + 1)|$
So how I proved it was simply using the ...
- 865
-3
votes
4
answers
109
views
why is the co-prime part not mentioned in the definition of the rational number?
Proving $\sqrt{2}$ an irrational number is a quite popular exercise, in precalculus courses, but if we look clearly the definition that is introduced, in the beginning of the course, it never ...
- 282
-1
votes
1
answer
52
views
Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same. [closed]
Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same.
If the assertion is not true then please prove ...
- 3
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 ...
0
votes
1
answer
137
views
proof of infimum and upper bound of $(1+1/n)^n$
I have to prove that 2 is the minimum, and therefore infimum of the set of all numbers $(1+1/n)^n$, where $n$ are positive integer numbers. And also that it is upper bounded, not necessarily to show ...
- 31
1
vote
0
answers
59
views
Exercise 6, Section 2.2 of Hoffman’s Linear Algebra
(a) Prove that the only subspaces of $\Bbb{R}$ are $\Bbb{R}$ and the zero subspace.
(b) Prove that a subspace of $\Bbb{R}^2$ is $\Bbb{R}^2$, or the zero subspace, or consist of all scalar multiples of ...
- 4,227
0
votes
1
answer
163
views
Prove that the closed interval [a, b] is compact.
I am studying the topology of $\mathbb{R}$ and I want to prove that the closed interval $[a,b]$ is compact using the Heine-Borel Theorem that a set in $\mathbb{R}$ is compact iff it is closed and ...
user1052225
0
votes
3
answers
179
views
Is a bounded polynomial constant? [duplicate]
I am trying this problem:
If $p(x)$ is a bounded polynomial for all $x\in \mathbb R$, then $p(x)$ must be a constant.
I am trying to prove it by contradition. So I assume that $p(x)$ is bounded for ...
- 290
3
votes
1
answer
94
views
Help With Basic Proof from Rudin PMA Chapter One
From Walter Rudin's Principles of Mathematical Analysis, Third Edition, page 20, Step 8:
I want to complete the proof of (b) by proving the following claim.
Claim: For $r\in{}\mathbb{Q}^+$ and $s\in{}...
- 515
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 ...
0
votes
1
answer
265
views
Proving trichotomy and transitivity from the definition of an ordered field
I'm reading a set of notes (and can provide the link if anyone is interested) which attempt to build up the properties of an ordered field. After defining a field based on its axioms, it defines an ...
- 1,418
0
votes
2
answers
105
views
Is there a proof for triangle inequality in $\mathbb{R}$ by contradiction/absurd?
I want to prove that given $a,b,c\in\mathbb{R}$ we have $|a+b|\leq|a|+|b|$ using an absurd and reaching a contradiction.
So, I state, by absurd, that $|a+b|>|a|+|b|$, but I can't reach the ...
- 369
0
votes
0
answers
137
views
Prove $(-a)^{-1} = -a^{-1}$
This is my proof for part (17) of Lemma 2.3.2 in Bloch's Real Analysis. I'd like if someone verifies that I did not miss or skip a step, did anything unjustified, or anything of this sort. I will ...
2
votes
0
answers
70
views
If $0<a<b$ and $0<c<d$ then $bd+ac>bc+ad.$ Can this inequality be generalised to include more than just two lots of two numbers?
$0.7<2.4,\ $ and $\ 0.8< 0.9.$ By calculation, we see that $0.7\times 0.8 + 2.4\times 0.9 > 0.7\times 0.9 + 2.4\times 0.8.$
Indeed, for any pair of two numbers $\ 0<a<b\ $ and $\ 0<c&...
- 20.6k
2
votes
1
answer
63
views
Proving a rational number is NOT in the lower Dedekind cut for a transcendental number
I'm attempting to argue with a finitist who claims that transcendental numbers can't be defined as Dedekind cuts without using an infinite predicate or in some other way requiring an infinite number ...
- 121
2
votes
1
answer
81
views
A slight anomaly in a question regarding real numbers
The leftmost digit of the decimal representations of the natural
numbers $2^n$ and $5^n$ is the same. Prove that such digit is equal to $3$.
I wanted to see some pattern in the given question(so as ...
0
votes
0
answers
30
views
Proof Outline beginning with basic properties of real numbers of an application of the Mean Value Theorem
I'm working on a project in Real Analysis and I am stuck on the following question:
Outline a proof, beginning with basic properties of the real numbers, of the following theorem - if f : [a,b] → (−∞,...
-3
votes
1
answer
75
views
How can I prove this question?
We know that it is not proved that $e^e$ is transcendental, so neither is the number that $e^{e\sqrt{2}}$. My question is, if one turns out to be, how can it be proved that the other is? Because there ...
user892441
0
votes
1
answer
37
views
Showing an inequality in the real positive numbers
I want to show that $\forall c>0$:
$$0<\left(1-\frac{(wl)^2}{r^2+(wl)^2}\right)^2+\left(\frac{wrl}{r^2+(wl)^2}\right)^2<(1-w^2lc)^2+(wrc)^2\tag1$$
Where all variables are bigger than zero.
...
- 23
-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: $$$$
If we have the two ...
0
votes
1
answer
364
views
Proof of Dedekind cuts.
This is my definition for Dedekind cuts:
A subset α of Q is said to be a cut if:
$α$ is not empty,$α\neq \mathbb{Q}$
If $p \in α,q\in\mathbb{Q}$,and $q<p$,then $q\inα$.
If $p\in α$,then $p<r$ ...
- 97
1
vote
0
answers
59
views
Real Analysis - Prove that there exists $n, m \in \Bbb{N}$ such that $2m\pi + \frac{\pi}{2} - \epsilon < n < 2m\pi + \frac{\pi}{2}$.
This is a claim that I had made while finding the supremum of $(\sin(n))_{n \in \Bbb{N}}$. The supremum is $1$ if there exists $n \in \Bbb{N}$ such that for all $\epsilon' > 0$, $1-\epsilon' < \...
- 4,335
3
votes
1
answer
67
views
How do I solve this in an understandable and direct way? [closed]
For each $i \in \Bbb N$, let $f_i: \Bbb N \mapsto \{0, 1\}$.
Let $A = \{f_i : i \in \Bbb N\}$ and $E = \{n \in \Bbb N : f_n(n) = 0\}$.
Does there exist a $f \in A$ such that $E = \{n \in\Bbb N : f(n) =...
user850289
3
votes
1
answer
154
views
Finding all real $(a,b,c)$ satisfying $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $a^2+b^2+c^2=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$
I have been trying to find my error in the following question for a while, but am yet to succeed:
Find all triples $(a,b,c)$ of real numbers that satisfy the system of equations:
$$\begin{align}
a+b+...
user814992
0
votes
4
answers
71
views
$Show: a\in \mathbb{Z} \Rightarrow 6\mid a^3 -a$
$Show: a\in \mathbb{Z} \Rightarrow 6\mid a^3 -a$
My attempt:
NTS: $6\mid a^3 -a$, so $(2\mid a^3 -a) \land (3\mid a^3 -a)$
Assume that $a\in \mathbb{Z}$, therefore I have 2 cases:
a is even $\...
- 129
0
votes
1
answer
35
views
Example of basis
Excuse me , can you see this question
, the collection of all open intervals (a,b) together with the one-point sets {n} for all positive and negative integers n is a base for a topology on a real ...
- 93
0
votes
3
answers
67
views
How do I prove that $\forall x\in\mathbb{R}:(\forall y\in\mathbb{R}: y>0 \implies 0\leq x\lt y)\implies x=0$
When reading this proof of the uniqueness of limits of sequences I stumbled across this argument (in the last few lines):
We have a $y\in\mathbb{R}$ with $y>0$. (Originally called $\epsilon$)
We ...
- 1
2
votes
1
answer
86
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 = ...
2
votes
1
answer
319
views
Bad Proof? Between any two reals is a rational number
I know about the proof found here: Proof there is a rational between any two reals.
I wanted to know if this similar proof is also correct?
Assume $x > 0$. Since $y > x$, it follows $y-x>0$....
1
vote
1
answer
63
views
When $ a^5 < 5 $ show that there exists b such that $ a<b, b^5<5 $
Here's my approach.
Since $ a^5 < 5, a<\sqrt[5]{5} $
By density of rational number, there must be integer $m$ and natural number $n$ such that
$ a< \frac{m}{n} < \sqrt[5]{5}$
If I let $...
- 119
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
2
votes
1
answer
278
views
Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
- 109
0
votes
3
answers
133
views
Prove using the axioms that $x>0$ implies $-x<0$
How to prove equations that if $x>0$, then $-x<0$ using the axioms of the real numbers $\Bbb{R}$ (if $x \in \Bbb{R}$)?
My university lecturer gave this as an exercise and I am stuck on which ...
- 109
0
votes
1
answer
113
views
How should I solve these inequalities?
These are two inequalities from my assignments. I don't know if it is too difficult or I am not so good at inequalities but please help me with full answers.
Let $a$,$b$,$c$ be three real positive ...
- 1,288
0
votes
4
answers
173
views
Prove that, there are 4 real roots of system of equations: $\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$
How can I prove that, there are 4 real roots of this system of equation?
Solve for real numbers:
$$\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$$
My attempts:
$$(7-x^2)^2+x=11 \Longrightarrow x^4 - ...
user548054
1
vote
1
answer
114
views
Prove that there exists an $N \in \mathbb{N}$ with $0 < N^{-1} < b-a$
Given are two numbers $a,b \in \mathbb{R}$ with $a<b$. Prove that there exists an $N \in \mathbb{N}$ with $N \geq 1$ with
$$0 < N^{-1} < b-a.$$
Show that there exists a $k \in \mathbb{Z}$ ...
- 345
1
vote
0
answers
67
views
Proof: For any subsequence $a_{n_k}$ Prove $\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n$
For any convergent subsequence $a_{n_k}$ of $a_n$, Prove: $$\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n.$$
My attempt
For this proof it should be noted that $...
- 70
0
votes
1
answer
24
views
Analysis and Compact intervals
Let $[a,b] \subseteq \mathbb{R}$ such that $a<b$ Then $\forall \epsilon>0$ $\exists$ $x_1,x_2$ $\in$ $[a,b]$ such that $|x_1-x_2|<\epsilon$
I would like hints on this particular problem, ...
user643073
-2
votes
2
answers
48
views
A natural number between two reals [closed]
How should I go about proving the following:
$\forall x \in \mathbb{R}, \exists n \in \mathbb{N}$
$ s.t. $
$20(3x^2 - 3x + 2) > 15n > 12(5x^2 - 5x + 2)$
- 319
0
votes
3
answers
108
views
Prove that if a $\neq$ 0 and a*b=a*d then b=d
This problem assumes that a, b, d $\in$ $\mathbb R$ with a $\neq$ 0. I've been trying to figure this out for a few days and I'm not even sure if I'm headed in the right direction with what I have so ...
user716286
0
votes
0
answers
42
views
Need a logical proof of [(c>0) and (|a|<c)] implies [(-c<a) and (a<c)]
I need a logical proof of the elementary statement about real numbers using order and field axioms
$((c>0)\wedge(|a|<c))\Rightarrow((-c<a)\wedge(a<c))$
- 131
0
votes
3
answers
295
views
Prove that $0< \frac{1}{2^{m}} <y$
If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$
I think I have to use Archimedean property to prove it. The Archimedean property ...
- 903
3
votes
1
answer
128
views
Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$
Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$.
Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$.
I don't have problems in proving $F\subseteq \mathcal{B}(\mathbb{R})\...
- 319
0
votes
0
answers
70
views
Proving the uniqueness of x=sqrt(r)
Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$
I would appreciate any nudge in the ...
-1
votes
1
answer
43
views
Proofs on inequalities of real numbers [closed]
So I have these inequalities (statements) to prove:
$x,y \in \mathbb{R}$
$\vert xy \vert \leq \frac{1}{2}(x^2 + y^2)$
$x,y \geq 0 \implies xy \leq \frac{1}{4}(x + y)^2$
I know that I have to use ...
0
votes
1
answer
40
views
Real Numbers Proof
working on proving:
Let $x,y \in \mathbb{R}$ be such that $x < y$. There exists $z \in \mathbb{R}$ such that $x < z < y$
This seems obvious but I'm having trouble using the properties of ...
0
votes
2
answers
110
views
Prove that $\forall \epsilon>0, \epsilon>a \implies 0 \geq a$
I am doing a course on basic real analysis in which firstly i am emphasising on real numbers. My book says that real number satisfies the following axioms.
1)Field Axiom
2)Extend Axiom
3)Order Axiom
...
- 792
1
vote
1
answer
113
views
Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$
Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$.
Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
- 5,154
1
vote
2
answers
318
views
Proof regarding indexed families of sets and intervals
I have the following problem:
Let $I$ be the set of real numbers that are greater than $0$. For each $x \in I$, let $A_x$ be the open interval $(0,x)$. Prove that $\cap_{x \in I} A_x = \emptyset$.
...
- 45
1
vote
1
answer
88
views
Deduce $\mathbb Q$ is a dense set $($in real numbers$)$
I am trying to deduce $\mathbb Q$ is a dense set $($in real numbers$)$ i.e. $x, y \in\mathbb Q$, there exists $\alpha$ in $\mathbb Q$ such that $x < \alpha < y$. I have let $x$ and $y$ be real ...
- 31