TITLE: There exists a rational sequence that converges to $\sqrt3$
QUESTION [6 upvotes]: I got a proof of this but I am quite sure that it is not what was expected on the exam. Also, this proof seems really kludgy and non-kosher.
Because of the density of the raitonals in the reals, there exists a $q\in\mathbb{Q}$ such that $\sqrt3-\frac1 n < q < \sqrt3$
For each n, let $a_n = q$.  ($\sqrt3-\frac1 n < a_n < \sqrt3$)
So for n=1. There exists a $q$ such that $\sqrt3-1 < q < \sqrt3$. Choose this q for $a_1$
For n=2, choose $q$ such that $\sqrt3-\frac1 2 < q < \sqrt3$.
Since $\sqrt3-\frac1 n$ converges to $\sqrt3$, $a_n$ must also converge to $\sqrt3$.
So we are constructing a sequence out of things that we are only know the existence of. Also does this require the axiom of choice?
Anyways, on the exam there was a hint "consider $S = \{r\in\mathbb{Q}|r>0 \,\mathrm{and}\,r^2<3\}$" And I am not exctly sure what to make of it other than the fact that $\sup S = \sqrt 3$
Edit: To be specific about my question, is this proof ok? And what is the standard proof (the proof that my prof was hinting at)?

REPLY [2 votes]: A simpler sequence to calculate than that using the continued fraction is to let $r_1=1/1$ and $r_2=4/2$, and generally if $r_n=p/q$ let $r_{n+1}=(p+3q)/(p+q)$. The first several terms:
$1/1,4/2,10/6,28/16,76/44,208/120,...$
The terms will be alternately below and above $\sqrt{3}$, and approach it. It's fairly fast, for example for the sixth term, $208/120-\sqrt{3}=.001285...$ It doesn't converge as quickly as does the sequence arising from the continued fraction. 
But this approach has the advantage that one can do any squareroot of a positive (nonsqure) integer $m$ this way. Start with $r_1=1/1$ and if $r_n=p/q$ let $r_{n+1}=(p+mq)/(p+q)$ Then the sequence $r_n$ will approach $\sqrt{m}$, the terms alternately below and above it.
As with the continued fraction method, there is a recursive formula going from one fraction in the sequence to the next; the recusion here looks a bit simpler than does the continued fraction one, in that one only need keep track of the current numerator and denominator.