I have the following problem:
Let $X$ be the set of all bounded sequences of real numbers with the metric
$d(\{x_n\},\{y_n\}) = \sup\{|x_n-y_n|: n \in \mathbb{N}\}$.
Which, if any, of the following subsets of $X$ is compact?
(1) The set $A$ of all convergent sequences
(2) The set $B$ of all sequences which converge to zero
(3) The set $C$ of all constant sequences
When I see this problem my first thought is that it is trivial, because for (1), (2), and (3) for each sequence in each set there will obviously be a convergent subsequence because the mother sequence is convergent. However, I'm not very familiar with working with sets of sequences, so am I actually going to be trying to figure out whether or not each sequence of sequences has a convergent subsequence? If so, how would I accomplish that?
Hint: The continuous image of a compact set is compact, and the map $f\colon X\to\mathbb R$, $\{x_n\}\mapsto x_1$ is continuous.