TITLE: An example of a bounded countably infinite subset of the real numbers.
QUESTION [9 upvotes]: I've been trying to think of an example of bounded, countably infinite subset of the real numbers.  However, knowing that countably infinite means can be put into 1-1 correspondence with the naturals, this doesn't seem intuitively obvious.
Thanks in advance.

REPLY [15 votes]: The set of rationals contained in $ [0,1] $ is another example.
Addendum
If you start off with any countably infinite set $ S \subseteq (- \infty,\infty) $ that is unbounded, then there is a quick way out. The inverse tangent function $ \tan^{-1} $ maps $ (- \infty,\infty) $ bijectively to the bounded interval $ \left( - \dfrac{\pi}{2},\dfrac{\pi}{2} \right) $, so the image $ {\tan^{-1}}[S] $ is a bounded and countably infinite set. :)

REPLY [10 votes]: $$\mathbb{Q} \cap [a,b]$$ where $a,b \in \mathbb{R}$, $a<b$.

REPLY [7 votes]: Let $A = \{ 1/n : n \in \mathbb{N} \}$ where $\mathbb{N} = \{ 1, 2, 3, ... \}$.
$A$ is bounded between 0 and 1, and has an obvious bijection with $\mathbb{N}$.

REPLY [5 votes]: $\left\{\frac{1}{n} | n \in \mathbb{N}\right\}$

It's bounded in $(0,1]$
It corresponds to $\mathbb{N}$ by $\varphi(n)=\frac{1}{n}$