Increasing Sequence of Rationals Let $x$ be any real number. Construct a sequence $x_n$ of rational numbers such that 
$$x = \sup\{x_n : n \in \mathbb{N} \}.$$
I was trying $x_n = [ 10^n x ]/10^n$, but is it actually monotone increasing?
If so, how to prove it analytically?
Thanks for any help.

Yes, your sequence $$x_n = \dfrac{\lfloor 10^n x \rfloor}{10^n}$$ will do the job. Your sequence is also monotone increasing.
HINT:
To prove monotone increasing, try to prove the following first:
$$a \lfloor x \rfloor \leq \lfloor ax \rfloor$$ where $a \in \mathbb{Z}^+$ and $x \geq 0$.
