I have the following definition:
Definition
A set A in a metric space (M, d) is said to be totally bounded if, given any $\epsilon >0$, there exist finitely many points $x_1, ..., x_n \in M$ such that $A \subset \bigcup_{i = 1}^{n} B_\epsilon(x_i)$.
Defintion:
The Hilbert cube $H^\infty$ is the collection of all real sequences $x = (x_n)$ with $|x_n|\leq 1$ for $ n = 1, 2, ...$
Furthermore the question refers to an exercise:
A metric space is called seperable if it contains a countable dense subset.
Question: How do I prove that $H^\infty$ is totally bounded?
I don't have any clue how I should handle this, thanks in advance!