Identity is of Exponential Order Epsilon

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Theorem

Let $I_\R: t \mapsto t$ be the identity mapping on $\R_{\ge 0}$.


Then $I_\R$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.

Proof

\(\ds e^{\epsilon t}\) \(\ge\) \(\ds 1 + \epsilon t\) Exponential of $t$ not less than $1 + t$
\(\ds \) \(>\) \(\ds \epsilon t\)
\(\ds \leadsto \ \ \) \(\ds K e^{\epsilon t}\) \(>\) \(\ds t\) $K = \dfrac 1 \epsilon$

$\blacksquare$