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Prove that $c_0 \subset B_0$
Suppose $\{c_n\} \subset c_0$. If $c_n \to x$ show that $x \in B_0$, where $B_0$ is the closed unit ball in $c_0$.
I want to start by showing that $x \in B_0$. This is where I am stuck, I need some kind of contradiction.
Any ideas?
A:
We have that $c_n \rightarrow x$ so $d(x,0) = \lim_{n \rightarrow \infty}d(x,c_n)$. By our hypothesis, this is bounded above by $\lim_{n \rightarrow \infty} d(0,c_n)$.
If $d(c_n,0) = \lim_{n \rightarrow \infty} d(0,c_n)$ for all $n$, we will have $d(x,0) = 0$ so $x \in B_0$.
However, we have that $d(c_n,0) = \lim_{i \rightarrow \infty}
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