Could anyone help me to prove this inequality? $$ \vert e^{-m} - e^{-n}\vert\leq \vert m - n\vert e^{-\min\left\lbrace m,n\right\rbrace} \qquad \mbox{ for all } m,n\geq 0 $$ I don't know how to start. Thank you!
2026-04-07 00:35:27.1775522127
Inequality with exponential function
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WLOG, $m>n$.
$$e^{-n}-e^{-m}=e^{-n}\left(1-e^{n-m}\right)<e^{-n}(m-n)$$
because $e^{t}\le1+t.$