Proof that all real numbers have a rational Cauchy sequence? I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for this fact? It can use other theorems from analysis, as long as they aren't too obscure.

Let $\alpha$ be any real number. 
Define 
$$a_n = \frac{\lfloor n \alpha \rfloor}{n}$$
where $\lfloor , \rfloor $ denotes the floor function.
Then $a_n \in \mathbb Q$. Moreover we have 
$$\frac{n \alpha -1}{n} \leq a_n \leq \alpha$$
and hence $a_n \to \alpha$.
