All Questions
Tagged with real-numbers analysis
217
questions
-1
votes
2
answers
87
views
Is there an open interval about a number that is $\textbf{not}$ $\pi$-rational?
Suppose I have an $x$ such that $x$ is not $\pi$-rational, i.e. $\frac{x}{\pi} \neq \frac{n}{m}$ where $n$ and $m$ are some integers. Does this mean that there is some open interval $I = (x - \epsilon,...
- 1,560
-1
votes
1
answer
32
views
Compute end points of interval $\{ x \in \mathbb R \mid a_i x + b_i = \max_j a_j x + b_j\}$. [closed]
Let $a_k,b_k \in \mathbb R$ for $k \in [4] := \{1,2,3,4\}$. Define $f_k:\mathbb R \to \mathbb R$ by $f(x) := a_kx+b_k$. For any $i \in [4]$, define $A_i := \{x \in \mathbb R \mid f_i(x) = \max_{k \in [...
- 9,575
-1
votes
2
answers
98
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How to show that $\lim_{x\to 0} (1+x)^{1/x}$ is same as $\lim_{x\to +\infty} (1+1/x)^{x}$?
I am trying to calculate $\lim_{x\to 0}(1+x)^{1/x}$ by using that $\lim_{x\to+\infty}(1+1/x)^x=e$.
By substitution with $a=\frac1x$ I get that $\lim_{x\to 0}(1+x)^{1/x}= \lim_{a\to +-\infty}(1+\...
- 13
-1
votes
2
answers
468
views
Countable and Uncountable Sets Question [closed]
I know and understand the basic theory behind what makes a set finite/infinite and countable/uncountable but I have no clue how to apply this to an actual question, my maths lectures are very theory ...
-1
votes
1
answer
57
views
Construction of real numbers, multiplication definition (Dedekind)
Definition:
Let $r,s∈ℝ$.
If $r>_ℝ0$ and ,$s>_ℝ0$ then the product of $r$ and $s$ is:
$r·_ℝs=\{p·_ℚq:p∈r\:and\:q∈s\}∪\{q∈ℚ:q≤_ℚ0\}$
If $r>_ℝ0$ and , $s≤_ℝ0$ then the product of $r$ and $s$ is:$...
- 686
-1
votes
1
answer
147
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How to show that Archimedes' principle is equivalent to 'for any c>0, $\exists$ k $\in$ $\Bbb N$ such that k - 1 $\le$ c $\lt$ k'
Archimedes principle: For any two positive real numbers M, $\varepsilon$, there exist a k in $\Bbb N$ such that M < k * $\varepsilon$.
How can I show the above statement is equivalent to the below ...
-2
votes
2
answers
88
views
Prove result of xy [closed]
If $$25^x = 7\quad \text{and}\quad 7^y = 125$$ then $xy=\frac{3}{2}$.
Can someone explain me why $xy$ is equal to $\frac{3}{2}$?
Thank you
