1
vote
3answers
37 views

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b> 0$ and $c\geq 0$ ? Is there a general proof for that relation ?
2
votes
1answer
67 views

Simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and a natural ...
5
votes
1answer
128 views

$\frac{\prod_{i=1}^n (1+x_i)-1}{\prod_{i=1}^n (1+x_i/\delta)-1} \stackrel{\text{?}}{\le} \frac{(1+x_n)^n-1}{(1+x_n/\delta)^n-1} $ .

Let $x_1 \le x_2 \le \cdots \le x_n$. Let $\delta>1$ be some positive real numbers. I assume that $0\le x_i <1$, for $i=1,\ldots,n$ and $x_n >0$. Does the following expression hold? $$ ...
6
votes
1answer
109 views

How to best understand Euclid's definition of equal ratios? How does it relate to Dedekind cuts?

This is something I've been wondering about. When I think of "ratios" $x/y$ and $z/w$ as being "equal", with $x$, $y$, $z$, and $w$ being real numbers, this means the results of dividing the real ...
5
votes
2answers
125 views

REVISTED$^2$: Fraction Existence Proof

Question 1: I'm asked to prove that there exists an $n\in\mathbb{N}$ such that $$\frac{1}{n+1}\leq\frac{a}{b}<\frac{1}{n},$$ where $0<\frac{a}{b}<1$. Here $\frac{a}{b}$ is a fraction in ...
2
votes
1answer
441 views

Nested Division in the Ceiling Function

During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that: $$ \left\lceil{\frac{n}{2^i}}\right \rceil= ...
1
vote
1answer
79 views

Looking for hints of this inequality

I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much! Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...