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Let $c_0$ be the sequences with $\lim_{n\rightarrow \infty} = 0$. Show that the closed unit ball $\{x\in c_0, \|x\| \leq 1\}$ is not compact in $\ell^\infty$.

I know a lemma that says that the infinite closed unit ball are not compact in infinite-dimensional normed spaces.

This seems strange to me. Does not all the subsequences of sequences in $c_0$ converges?

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    doesnt we need to consider the norm?2012-12-07

2 Answers 2

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Let $e_n$ be $0$ except for a $1$ in the $n$th coordinate. Clearly $e_n \in c_0$. Then $\|e_n\|_\infty = 1$, but $\|e_n -e_m \|_\infty = 1$ whenever $n\neq m$. Hence $e_n$ can have no convergent subsequence, hence the set $\{ x \in c_0 | \|x\|_\infty \leq 1 \}$ cannot be compact.

And yes, all subsequences of $c_0$ must converge, since any subsequence of a convergent sequence must converge to the same limit.

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    $e_n$ is zero except at index n. Hence $\lim_{i \to \infty} [e_n]_i = 0$. Hence $e_n \in c_0$. I do not understand what you mean by 'how do we know that this sequence are going on forever'. I have defined it for all $n$, so it must go on for all $n \in \mathbb{N}$.2012-12-07
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Define $e_n$ to be the sequence that is zero everywhere except at $n$ where its one. Then $||e_n||=1$ and if we consider the sequence $\{e_n\}_{n=1}^\infty$ then I claim that it has no convergent subsequence. In particular notice that the sequence converges pointwise to $0$, so any limit would have to be $0$. But we have that $||e_n||=1$ so it can't converge in norm to $0$. Thereby it has no convergent subsequence.

Notice that the question is asking about a convergent subsequence of a sequence of sequences. Not a convergent subsequence of an element of $C_0$.