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The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant$C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of$(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by$C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form$U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and thepositive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ issharp but the values and asymptotic behavior of the optimal constants remain amystery. The first estimates for these constants had exponential growth. Veryrecently, a new panorama emerged and the importance, for many applications, ofthe knowledge of the optimal constants (denoted by $(K_{n})_{n=1}^{\infty}$)was stressed. The title of this paper is part of our Fundamental Lemma, one ofthe novelties presented here. It brings surprising new (and precise)information on the optimal constants (for both real and complex scalars). Forinstance, [K_{n+1}-K_{n}<\frac{0.87}{n^{0.473}}] for infinitely many $n$'s. Inthe case of complex scalars we present a curious formula, where $\pi,e$ and thefamous Euler--Mascheroni constant $\gamma$ appear together:[K_{n}<1+(\frac{4}{\sqrt{\pi}}(1-e^{\gamma/2-1/2}){\sum\limits_{j=1}^{n-1}}j^{^{\log_{2}(e^{-\gamma/2+1/2}) -1}%})] for all$n\geq2$. Numerically, the above formula shows a surprising low growth,[K_{n}<1.41(n-1)^{0.305}-0.04] for every integer $n \geq2$. We also provide abrief discussion on the interplay between the Kahane-Salem-Zygmund and theBohnenblust-Hille (polynomial and multilinear) inequalities.

There exist multilinear Bohnenblust-Hille constants $(C_{n})_{n=1}^{\infty}$ with $\displaystyle \lim_{n\rightarrow \infty}(C_{n+1}-C_{n}) =0.$