I need to give an example of a closed bounded subset of $l_\infty$ (the collection of all bounded real sequences) that is not totally bounded.
Question: How do I do this? What would a subset of a collection like $l_\infty$ look like? Would this be a collection of bounded real sequences, or would this be a subset of the range of all the bounded real sequences?
Just so that you have an answer here (but this is just the same answer to which Math1000 posted a link in the comments): for each $n\in\mathbb N$, define $e(n)\in l^\infty$ by$$e(n)_k=\begin{cases}1&\text{ if }n=k\\0&\text{ otherwise.}\end{cases}$$The the set $E=\{e(n)\,|\,n\in\mathbb N\}$ is bounded, being a subset of the closed ball centered at $0$ with radius $1$. But the distance between any two distinct elements of $E$ is $1$. Therefore, there is no finite union of open balls with radius $\frac12$ containing $E$.