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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!

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    What is an $\varepsilon$-linear space? And your functions $f_n$ are not elements of $C^\infty(\mathbb{R})$?2012-05-22
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    @Vobo it seems a safe bet that $\varepsilon$ is a replacement for $\mathcal{E}$ and that $\mathbb{R}$ denotes the domain, not the range. .@Toby: if my edit does not reflect your question, please add some clarifications.2012-05-23
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    @t.b.: Oh sure, now I would not hold against you.2012-05-23

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