Let $\mathcal{E}$ be the space $C ^{\infty}(\mathbb R)$ with the system of seminorms: $ p_{N,n}(f) := \max{\lbrace |f^{(k)}(t)| : k = 0, 1, \dots , n; t \in [-N, N] \rbrace},\quad n = 0, 1, 2, \dots; N = 1, 2, \dots. $
So, I have to find the limit of $f_n(t) = \dfrac{1}{t + n + i/n}$ in the space $\mathcal{E}$.
I understand, that it is 0, but I don't know, how to prove that it exists.
Thank you!