How do I know that a continued fraction of an irrational number converges?

Question

I am familiar with the algorithm to calculate the continued fraction of an irrational number, but I was wondering how we can be sure that the fraction actually converges to the irrational we are concerned with. I know how to show the convergence of particular continued fractions, but I don't see how we could do this with a general fraction. I've seen that any finite continued fraction of an irrational is just a matter of algebra, but how can we be sure that repeatedly applying that algorithm gives us something that converges to the relevant irrational? If it helps in answering the question, I am interested in particular in the convergence of continued fractions of surds $\sqrt{n}$, but I thought I should ask for the general case. If it's considerably easer to show convergence for surds then it may be worth separating the cases.

Answer 1

You'll find proofs of the convergence in any decent exposition of continued fractions (Chapter X of Hardy and Wright's An Introduction to the Theory of Numbers, for example).

In particular, if $\ \left\{\frac{p_n}{q_n}\right\}_{n=0}^\infty\ $ are the convergents of the continued fraction expansion of any real number $\ x\ $, where $\ p_n,q_n\ $ are relatively prime integers for each $\ n\ $, and $\ p,q\ $ are any integers with $\ 1\le q\le q_n\ $, then $$ \left|x-\frac{p_n}{q_n}\right|\le\left|x-\frac{p}{q}\right|\ . $$ That is, of all rational approximations to $\ x\ $ with denominators not exceeding $\ q_n\ $ in magnitude,$\ \frac{p_n}{q_n}\ $ is the best. This is Theorem $181$ on p.$151$ of Hardy and Wright's above-cited text, for example. Since it follows from the recursion $\ q_n=$$\,a_nq_{n-1}+$$\,q_{n-2}\ $ (with $\ a_0,a_1,\dots\ $ being the partial quotients) that $\ q_n\rightarrow\infty\ $, then it obviously follows from this theorem, and the density of the rationals in the reals, that $\ \frac{p_n}{q_n}\ $ must converge to $\ x\ $ as $\ n\rightarrow\infty\ $.