All Questions
Tagged with real-numbers proof-writing
127
questions
2
votes
1
answer
87
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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 $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
- 113
2
votes
1
answer
322
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Bad Proof? Between any two reals is a rational number
I know about the proof found here: Proof there is a rational between any two reals.
I wanted to know if this similar proof is also correct?
Assume $x > 0$. Since $y > x$, it follows $y-x>0$....
- 3,422
1
vote
1
answer
63
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When $ a^5 < 5 $ show that there exists b such that $ a<b, b^5<5 $
Here's my approach.
Since $ a^5 < 5, a<\sqrt[5]{5} $
By density of rational number, there must be integer $m$ and natural number $n$ such that
$ a< \frac{m}{n} < \sqrt[5]{5}$
If I let $...
- 195k
4
votes
2
answers
172
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Proving the density of a function in an interval.
I am reading Steven G. Krantz's Real Analysis and Foundations and came across this problem.
Problem: Let $\lambda$ be a positive irrational real number. If $n$ is a positive integer, choose by the ...
- 93.3k
0
votes
3
answers
3k
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Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
(1) I start with an obvious fact:$$0 = 0$$
(2) Add $(-1)$ to both sides of the equation:
$$0 + (-1) = 0+ (-1)$$
(3) Zero is the ...
- 2,437
2
votes
1
answer
281
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Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
- 70.3k
0
votes
3
answers
133
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Prove using the axioms that $x>0$ implies $-x<0$
How to prove equations that if $x>0$, then $-x<0$ using the axioms of the real numbers $\Bbb{R}$ (if $x \in \Bbb{R}$)?
My university lecturer gave this as an exercise and I am stuck on which ...
- 43
0
votes
1
answer
113
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How should I solve these inequalities?
These are two inequalities from my assignments. I don't know if it is too difficult or I am not so good at inequalities but please help me with full answers.
Let $a$,$b$,$c$ be three real positive ...
- 195k
5
votes
1
answer
6k
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For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?
If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$?
Please help me to clarify the above. Thanks in advance.
- 4,246
0
votes
4
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173
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Prove that, there are 4 real roots of system of equations: $\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$
How can I prove that, there are 4 real roots of this system of equation?
Solve for real numbers:
$$\begin{cases} y^2+x=11 \\ x^2+y=7 \end{cases}$$
My attempts:
$$(7-x^2)^2+x=11 \Longrightarrow x^4 - ...
CommunityBot
- 1
1
vote
1
answer
114
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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 $k \in \mathbb{Z}$ ...
- 610
4
votes
2
answers
12k
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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 ...
CommunityBot
- 1
1
vote
0
answers
67
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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
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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, ...
- 68.8k
-2
votes
2
answers
48
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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. $
$20(3x^2 - 3x + 2) > 15n > 12(5x^2 - 5x + 2)$
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