# Can I approximate real numbers by rational numbers monotonically?

Really the title says it all: I know that for any real number $x$ there exists a sequence of rationals whose limit is $x$, but I don't know if such a sequence can be taken to be monotonically increasing or decreasing. I want to know because it would provide me with a quick solution to a theoretic problem I have. Thank you.

Edit: Along with a yes or a no, I'd appreciate justification.

• "monotonic," not "monotonous." – symplectomorphic May 1 '14 at 16:44

Yes your sequence can be taken to be monotonically increasing or decreasing. To see this, just note that for any $|q_n - x| \geq \epsilon > 0$, you can find $q_{n+j}$ which is closer than $q_n$ to $x$ for all sufficiently large $j$, if $q_n \to x$. Then just use the fact that if you have an infinite sequence that approaches $x$, then infinitely many terms in the sequence are $\geq x$ or infinitely many terms are $\leq x$, and you're done.
Yes, it can be monotonic. For increasing, take rational $a \lt x$ At each step, find a rational between the last value and $x$ The sequence will be increasing.