How can a set be bounded and countably infinite at the same time? There is a theorem in my textbook that states,

Let $E$ be a bounded measurable set of real numbers. Suppose there is a bounded countably infinite set of real numbers $\Lambda$ for which the collection of translates of $E$, $\{\lambda + E\}_{\lambda \in \Lambda}$, is disjoint. Then $m(E) = 0$.

I'm a little confused about this theorem, because it's saying that a set is bounded and countably infinite at the same time. But if a set is bounded, isn't it supposed to be finite?
Thanks in advance 

No: $[0,1]$ is a bounded set that isn’t even countable, let alone finite. An unbounded set of real numbers is necessarily infinite, but a bounded set can be of any size up to and including the cardinality of the entire set of real numbers.
To prove this, use the well-known fact, proved in several places on this site, that there is a bijection $\varphi:\Bbb R\to(0,1)$. Let $A$ be any subset of $\Bbb R$; then $\varphi[A]$ is a subset of $(0,1)$ of exactly the same cardinality, and every subset of $(0,1)$ is bounded.
