Bound for the quasinorm of sum of functions 
Let $\|\cdot \|$ be a quasinorm on functions, thus $\|cf\|=|c|\|f\|$ for scalar $c$, $\|f\|=0$ if and only if $f=0$, and we have the quasitriangle inequality
  $$\|f+g\|\lesssim\|f\|+\|g\|\ \ \ \ \ (1)$$
  for all functions $f,g$. Let $f_n$, $n=1,2\dots,N$ be a sequence of functions obeying the bounds
  $$\|f_n\|\lesssim 2^{-\varepsilon n}$$
  for some $\varepsilon>0$. Prove that
  $$\|\sum_{n=1}^N f_n\|\lesssim_\varepsilon 1$$

We denote the implied constant in $(1)$ by $C$. Then from (1), we have that
$$\|\sum_{n=1}^N f_n\|\leq \sum_{n=1}^N C^n\|f_n\|\leq \sum_{n=1}^N C^n 2^{-\varepsilon n}.$$
I don't know how to deal with the case $C\geq 2^{\varepsilon}$. The hint says we can use (1) to reduce the case where $\varepsilon$ is large.
 A: Let's prove that
$$
\| \sum_{n=1}^N f_n \| \leq D_\varepsilon
$$
for some constant $D_\varepsilon$ dependent on $\varepsilon$. We have $D_\varepsilon$ such that the above inequality works when $\varepsilon$ is very large compared to $C$, as you have worked out. Say we have defined $D_{\varepsilon}$ for $\varepsilon \geq M$. Now let $\varepsilon \in [M/2, M)$. We will reduce to the case of large $\varepsilon$. WLOG put $N = \infty$ by extending the sequence with $f_n = 0$ for $n > N$.
Split the sequence $(f_n)$ into subsequences $(f_{2n})$ and $(f_{2n+1})$. We have $g_n = f_{2n}$ is a sequence with $\| g_n \| \lesssim 2^{-2\varepsilon n}$ and $h_n = f_{2n+1}$ has $\| h_n \| \lesssim 2^{-2 \varepsilon n}$ as well. Since $2 \varepsilon \geq M$ is a value we have the result for, we have
$$
\| \sum_{n=1}^\infty f_n \| \leq C \| \sum_{n=1}^\infty f_{2n} \| + C \| \sum_{n=1}^\infty f_{2n+1} \| \leq 2CD_{2\varepsilon}.
$$
Therefore, we may put $D_\varepsilon = 2CD_{2\varepsilon}$, and we have defined $D_\varepsilon$ for all $\varepsilon \geq M/2$. By repeating this process to define the constant for all $\varepsilon \geq M/4$, then $M/8$, etc., we get the desired inequality.
