Counterexample to Bolzano-Weierstrass in infinite dimension From Bolzano-Weirerstrass we can demonstrate that in a normed vector space $E$ of finite dimension, every bounded sequence admits a limit point.
What are some counterexamples in infinite dimension? Does there exist a counterexample in every infinite dimensional normed space?
I believe this one works: Let $E$ be the space of sequences of real numbers with finite support, equipped with the norm $\| (a_k)_{k \in  \mathbb{N} } \|=\sup |a_k|$. Then take define the sequence $(s_n)$ as follows: $s_n$ is the sequence whose $n$th term is $1$ and every other term is $0$. Then $(s_n)$ is bounded, and we can easily show that it has no limit point.

Yes, that's the obvious counter example. You don't need to require finite support, when using supremum norm you only need the sequences to be bounded. Let the $j$th sequence be $k\to \delta_{jk}$ where $\delta_{jk}$ is the  Kronecker delta. Every element in the sequence has norm $1$ and the distance between any two elements is $1$ so clearly this is a bounded sequence that and no subsequence is Cauchy and therefore not convergent.
