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Let $c$ be the space of all sequences that converge in $(\mathbb F,|\cdot|)$ where $\mathbb F$ is either $\mathbb R$ or $\mathbb C$.

Endow $c$ with the norm $\|x\|=\sup_{n\in\mathbb N}|x_n|$. I am able to show that this defines a norm, but how can I show that this norm is well-defined? That is, how can I show that $\|x\|$ is finite for all $x\in c$?

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It doesn't make sense to say that you can show that this defines a norm but not that the norm is well-defined. There is no such thing as defining a norm that isn't well-defined. –  joriki Sep 19 '12 at 16:48
    
A convergent sequence is bounded. –  copper.hat Sep 19 '12 at 16:56
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If $x \rightarrow c$, then for some large enough $N$, we have that $|x_n - c| < 1$ for all $n \ge N$. So $\sup_{n \ge N} |x_n| \le |c| + 1 < \infty$. Also $\sup_{n < N} |x_n| < \infty$ since there are only finitely many terms here. So $\sup_{n\in \mathbb{N}} | x_n| < \infty$.

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On the other hand, if a sequence is such that $\sup_n |x_n| = \infty$ then for every $N \in \mathbb N$ there exists $n$ such that $x_n > N$. But then $x_n$ cannot converge to any $L \in \mathbb F$, hence your norm must be well-defined since the sup is finite on all convergent sequences.

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