# Tagged Questions

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### 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 ?
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### Existence of a 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 for every ...
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? $$... 1answer 127 views ### How to best understand Euclid's definition of equal ratios? How does it relate to Dedekind cuts? This is something I've been wondering about. When I think of "ratios" x/y and z/w as being "equal", with x, y, z, and w being real numbers, this means the results of dividing the real ... 2answers 129 views ### REVISTED^2: Fraction Existence Proof Question 1: I'm asked to prove that there exists an n\in\mathbb{N} such that$$\frac{1}{n+1}\leq\frac{a}{b}<\frac{1}{n},$$where 0<\frac{a}{b}<1. Here \frac{a}{b} is a fraction in ... 1answer 480 views ### Nested Division in the Ceiling Function During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that:$$ \left\lceil{\frac{n}{2^i}}\right \rceil= ...
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 ...