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 $x\in c$?