I'm trying to prove the following : $\forall N\ge 1$ $ : \exists (P_n^{(N)})\in (\mathbb R[X])^{\mathbb N}$ : $ ||e^{-t}P_n^{(N)}(t)-e^{-Nt}||_{\infty,\mathbb R+}$ $ \rightarrow 0$ as $n\rightarrow +\infty$
For $N=1$ one can take $P_n^{(1)}=1$, for $N=2$ one can show that $P_n^{(2)}(t)=\displaystyle\sum_{m=0}^n \dfrac{(-t)^m}{m!}$ will work. I'm trying to prove it by induction on $N$.
I guess it is natural to try to define $P_n^{(N+1)}=P_n^{(2)}P_n^{(N)}$ but I can't prove the uniform convergence since $P_n^{(2)}$ is not bounded in uniform norm.