Construct a sequence with certain property

Question

Construct a sequence $(s_n)$ which satisfies the following property: $\forall x \in \mathbb{R}$ and $\epsilon > 0$, there exists some $N$ such that $|x−s_N| < \epsilon$

Answer 1

To expand on what Seth said, the rationals are countable, so we can have a sequence running over them. They are also dense in the reals. So for any rational point $q$, and for any $\epsilon\gt 0$ there is a real point $r$ such that $|q-r|\lt\epsilon$

So if we take the sequence $s_n$ running over the rationals, $\forall x\in\mathbb{R}$ there is an $N$ such that $|x-s_N|\lt\epsilon$

Answer 2

Ellya's answer is probably the simplest answer to understand, but a slightly different approach is to construct a single formula that yields a dense sequence. For example, let

$$f(x)=\sqrt x\sin(\log x)\qquad{\rm and}\qquad x_n=f(n).$$

Then this is dense because $f(x)$ has range $\Bbb R$ (even when restricted to $(a,\infty)$ for some $a$) and the derivative goes to $0$.