I want to prove the following :
Let $f:\mathbb{R}^+ \to \mathbb{R^n}$ and $C,a,b \in \mathbb{R}^+$ with $a<b$ then $\exists C'$ s.t. $\forall t \geq0$ :
$$||f(t)|| \leq C (1+t)^{n-1}e^{-bt} \implies ||f(t)|| \leq C'e^{-at}.$$
I have proven that :
- $C (1+t)^{n-1}e^{-bt}$ vanishes at infinity for all n,
- Then $C (1+t)^{n-1}e^{-bt}$ is bounded (on $\mathbb{R}^+$),
- $\forall m, C (1+t)^{n-1}e^{-bt}=o_{t \to \infty}(t^m)$.
Can I deduce the result from this ? Thank you.
Edit
I think I have answered myself, I went the wrong way.
The following quantity : $$\dfrac {C (1+t)^{n-1}e^{-bt}}{e^{-at}}=C (1+t)^{n-1}e^{-(b-a)t}$$ is bounded (on $\mathbb{R}^+$) by a certain quantity $M$ (because it vanishes at infinity and is continuous on $\mathbb{R}^+$).
Then it follows immediately, $\forall t>0$ : $$||f(t)|| \leq C (1+t)^{n-1}e^{-bt} \leq Me^{-at}.$$
Is it ok ?