All Questions
Tagged with real-numbers proof-writing
127
questions
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 ...
