I often see the following inequality is used over and over again $ 1−x⩽e^{−x} $ for $x \in \mathbb{R}$, for proving or deriving various statements.
As a layman, I haven't seen this inequality appearing in any class I have taken in my life. So it seems quite unnatural to me, and seems just a special result for linear function and exponential function. It is not yet part of my instinct to use it for solving problems. So I want to fill up this indescribable gap within my knowledge.
I wonder if there are other similar results for possibly other commonly seen functions (elementary functions?).
Is there some source listing such results?
Thanks and regards!