TITLE: Is it right that the fundamental recurrence of an arbitrary continued fraction cannot be proved without induction?
QUESTION [8 upvotes]: Let $\dfrac{A_{n}}{B_{n}}$ be the $n^{th}$ convergent (approximant)
$$

\frac{A_{n}}{B_{n}}=b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{\begin{array}{c}
b_{3}+ \\ 
\\ 
\end{array}
\begin{array}{cc}
\ddots  &  \\ 
& 
\end{array}
+\dfrac{a_{n-1}}{b_{n-1}+\dfrac{a_{n}}{b_{n}}}}}}
$$
of a continued fraction. $A_{n}$, $A_{n-1}$, and $A_{n-2}$ satisfy the
recurrence
$$
\begin{eqnarray*}
\begin{pmatrix}
A_{n} \\ 
B_{n}
\end{pmatrix}
&=&
\begin{pmatrix}
A_{n-1} & A_{n-2} \\ 
B_{n-1} & B_{n-2}
\end{pmatrix}
\begin{pmatrix}
b_{n} \\ 
a_{n}
\end{pmatrix}
\quad \text{for }n\geq 1, \\
&& \\
\text{and }
\begin{pmatrix}
A_{-1} & A_{0} \\ 
B_{-1} & B_{0}
\end{pmatrix}
&=&
\begin{pmatrix}
1 & b_{0} \\ 
0 & 1
\end{pmatrix}
,
\end{eqnarray*}
$$
which may be proved by induction.
Question: Is there a non inductive proof? 
EDIT: I changed the title. Based on the comment below by anon

The numerators and denominators are defined inductively - and I don't
  believe you'll find an explicit formula for the n-th convergent of an
  arbitrary continued fraction (though maybe of particular ones with
  specific patterns, like $\varphi$) - so I don't see any other way to
  say prove anything about the convergents except inductively.

I reformulate the question to
Question: Is it right that the fundamental recurrence of an arbitrary continued fraction cannot be proved without using mathematical induction?

REPLY [3 votes]: A continued fraction may be defined

as a composition of linear fractional transformations;
as an expression of the form
$$b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\, \quad }}}}$$
by the fundamental recurrence itself.

In both cases 1 and  2 the recurrence relation is proved by induction, as far as I have seen, namely in recent books.
For a generic continued fraction I now think there is no other way of deriving the fundamental recurrence either from 1 or 2.