1. ## Subsequences

Give an example of a sequence that has subsequences converging to every real number
in [0; 1] and no other real numbers.

I am confused about subsequences. If I created a sequence that simply listed decimal expansions like this:
0, 1, .1, .2, .3..., .9, .01, .02, ....09,...
would that be sufficient?

2. ## Re: Subsequences

Yes, it works. Maybe you should give a formal argument.
An other example is given by the sequence $\displaystyle \left\{|\sin n|\right\}$.

3. ## Re: Subsequences

Ahh. That's much nicer. Thanks.