I was reading page 12 of this note on the internet, but it seems unclear to me why the following inequality holds:
from the 2nd to 3rd line of the above image.
All I could argue is that $e^x-1 = (e^{x/2}-1)(e^{x/2}+1) $, hence, we have that $$ (e^x-1)^{-1} \le (e^{x/2}+1)^{-1} .$$
where $x$ is such that $e^{x/2}-1 \ge 1 \Leftrightarrow x \ge \log 4. $ But this is no where as good as the inequality provided... I guess we split into cases? As this is part of a proof for DCT, so we need a function that dominates $g_N$.