I am attempting Q1 in Tao's lecture notes on harmonic analysis.
Let $\|\|$ be a quasi-norm and let $f_n$, $n = 1, 2, . . . , N$ be a sequence of functions obeying the bounds $\|f_n\| \lesssim 2^{-\epsilon n}$ for some $\epsilon > 0$. Show that $\|\sum\limits_{n=1}^N f_n\| \lesssim_\epsilon 1$.
If $\epsilon > \log_2(C)$ where C is the constant in the quasi-triangle inequality, you can easily just iterate and then apply the given bound term-wise to get a geometric sum. However, I don't see what to do if $\epsilon$ is small.
The notes claim you can use the quasi-triangle inequality to reduce to the case where $\epsilon$ is large but I don't see it.
Let $k\in\mathbb{N}$ be such that $\epsilon k > \log_2(C)$, that is $\epsilon k$ is the kind of large $\epsilon$ we'd like to have. Split the sequence into subsequences such as $$ (f_1, f_{1+k}, f_{1+2k}, f_{1+3k}, \dots ) \quad \text{and}\quad (f_2, f_{2+k}, f_{2+2k}, f_{2+3k}, \dots) $$ For each of these subsequences we in effect have $k\epsilon $ in the exponent, so the backward summation works to estimate the sum by $\lesssim 1$. And the number of these subsequences depends only on $\epsilon$ and $C$ (not on $N$), so just add them up and the effect of this summation goes into the implicit constant in the inequality.