# Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

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### What is the minimum value of the expression? [duplicate]

Given reals $x_i \ge 0$ for $i=1,2,3,...,2n$ and $\sum_{i=1}^{2n}x_i=1$ find the minimum value of $\sum_{i=1}^{n}x_i+\sum_{i=n+1}^{2n}{x_i}^2$. I tried the case $n=1$ and got the value $\frac{3}{4}$. ...
1 vote
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### Necessity of universal quantifier to represent a theorem with logic symbols

I have a preference to reduce the proof steps of a theorem, and the theorem itself, into logic symbols as much as possible. Not just because it is aesthetically appealing, but because it makes makes ...
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### Conditions that a sequence should satisfy to be an eventually monotone sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that: $a_n\in[0,1]$, $\forall n\in\mathbb{N}$ $\lim_{n\to\infty}a_n = 0$ $\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$ $a_{n+1} \leq a_n$...
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+100

### Additive real semigroups with an unbounded below part

Look at $$(*)\;\;\; H=\{ m\sqrt{2}+k|m,k\in \mathbb{Z}, m\geq 2\}\cup (\mathbb{Z}_++\sqrt{2}).$$ It is obvious that $H$ is an additive sub-semigroup of real numbers. But, it is interesting to know ...
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### How to more formally prove this inequality

This is a simple problem I came up with while doing another problem: Given: $n < (n + \frac{1}{2}) < y < (n + 1)$ Prove: $|y - n| > |y - (n + 1)|$ So how I proved it was simply using the ...
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1 vote
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### why is $L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\}$ not the set of all Dedekind cuts?

Let the set $L$ be definded as $$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$ where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$. Clearly, every member of $L$ is neither an empty set ...
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### Prenex Normal Form of a Simple Proposition Reads Strangely.

I was trying to convert a simple (true) proposition concerning the real numbers to Prenex Normal Form but arrived at a logical statement that didn't appear equivalent to what I started with. The ...
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### Consider intervals of real numbers (a,b) and (c,d) with a<c. Is their intersection open? [closed]

Consider intervals of real numbers (a,b) and (c,d) with a<c. Then: A) The intervals are disjoint B) Their intersection is not empty C) Their intersection is closed D) Their intersection is open It ...
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