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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 $C^\infty(\mathbb{R})$? – Vobo May 22 '12 at 23:11
@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. – t.b. May 23 '12 at 9:08
@t.b.: Oh sure, now I would not hold against you. – Vobo May 23 '12 at 10:14

In a locally convex space $X$ with seminorms $\{p_k | k \in I\}$ where $I$ denotes an index set, a sequence (or even more general a net) $(x_n)_n$ of elements of $X$ converges to some $x \in X$, iff for each $k \in I$ the real sequence (net) $(p_k(x_n - x))_n$ converges to 0.

Applying this to your situation, fix an $N$, take some $k$, and check $|f^{(j)}_n|$ for $n\to\infty$: For $n>2N+1$ and $t\in [-N,N]$, $|f^{(j)}_n(t)| <= (j!+1)/n$, hence $p_{N,k}(f_n) \to 0$ as $n\to\infty$.

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