Given $A \subset \mathbb{R}$ is bounded, what is an open cover of $A$?

Question

I am looking for a good example of a covering of a bounded set $A$ in $\mathbb{R}$

Currently my example is $\{I_n\} = \{(k + n - \frac{\epsilon}{2^n}, k + n - \frac{\epsilon}{2^n})\}_{n \geq 0}$, where $k = \inf(A)$, $\epsilon > 0$

Are there much more generic or clever examples?

Answer 1

Begin with a fishing line of arbitrary but finite length. Proceed to cover it with a collection of pancakes whose edges have been removed. Observe that it takes a finite number of pancakes to cover the fishing line. Eat the pancakes. Enjoy math.

Answer 2

Put $I_k=(-k,k)$ for all $k\in\mathbb {N}$. Because $A$ is bounded, exist $k_0\in\mathbb {N}$ such that $A\subset (-k_0,k_0)$ and $\{I_k\}_{1\leq k\leq k_0}$ is a covering by open sets for $A$