MathOverflow

I search to prove ($n\in \mathbb{N}_0$)
$$ |f_n(x)| \le 1 \text{ for } x\in[0,1]$$
where

When studying motivic homotopy theory, I find that using language of $\infty$-categories often simplifies a lot, comparing to the traditional method using model categories. Although I’m familiar with model categories now, I have only vague notions about...

Conjecture: Let $\Omega(n)$ be the total number of prime factors of $n$ (with multiplicity). For $k \ge 1$, let $a, b$ be consecutive integers such that $\Omega(a)=\Omega(b)=k$.I propose the inequality: $b^{\frac{1}{k+1}} - a^{\frac{1}{k+1}} < \frac...

Let $F:C_1\to C_2$ be an equivalence of categories. Suppose that $C_1, C_2$ are both exact categories and $F$ sends short exact sequences (SES) in $C_1$ to SES in $C_2$. Let $G$ be the quasi-inverse of $F$.
Question: Are there any conditions we can put on $F$ that ensures that $G$ sends SES in $C_2$ to SES in $C_1$? Or what can we say about $G$ regarding SES? (Or how do you check exactness in the two examples in the motivation below?)...