[Community Question] Algebra-Precalculus: why is $1/n < x < 1/(n+1) \leftrightarrow n+1 < 1/x < n $ this an equivalence
One of our user asked:
$1/n < x < 1/(n+1) \leftrightarrow n+1 < 1/x < n $ ? could someone explain me why this is true ?
One of our user asked: Is there an explicit example of an additive map $\mathbb{R}^n \rightarrow \mathbb{R}^m$ which is not linear? (I have mostly thought about the question when $m = n = 1$ , and I don't think the general case is any easier.) I know that something like $f: \mathbb{C} \rightarrow \mathbb{C}$ which sends $f: z \mapsto \text{Real}(z)$ would be additive but not $\mathbb{C}$ -linear. I also know that since $\mathbb{R}$ is a $\mathbb{Q}$ -vector space, I can find some example where $1 \mapsto 1$ and $\sqrt{2} \mapsto 0$ . Is there an explicit example? By explicit, I mean, given an element in the domain, there would be some procedure to decide where it maps. Thank you!
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