Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a.

Question

Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a.

I've been stumped for days, my guess is that it is an addition of sequences each expressing its own decimal place, but I can't figure out how to make that consider all possibilities.

Answer 1

Hint:

$\mathbb Q$ is a countable set, meaning that there exists a bijection between $\mathbb N$ and $\mathbb Q$. Also, $\mathbb Q$ is dense in $\mathbb R$.

Answer 2

Hint:

$0, 0.1, 0.2, 0.3, \ldots, 0.9, 0.01, 0.02, 0.03, \ldots 0.09, 0.11, 0.12, \ldots, 0.19, \ldots 0.99, 0.001, 0.002, \ldots$

It's somewhat simpler in binary:

$0, 0.1, 0.01, 0.11, 0.001, 0.011, 0.101, 0.111, \ldots$

Answer 3

One possible sequence :

$$S = (0,0,\frac{1}{2}, 0, \frac{1}{4},\frac{2}{4}, \frac{3}{4} , \cdots ) $$

We can write this as

$$S_{2^n+k} = \frac{k}{2^n}$$

for $0 \leq k< 2^n$