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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.2012-09-19
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    A convergent sequence is bounded.2012-09-19

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