Linear Fractional Transformation of Continued Fractions with bounded Partial Quotients: Arithmetical and Geometrical Point of View

In this paper I will give new proofs of some simple theorems concerning continued fractions. Like J.O. Shallit said, "the proofs in the literature seem to be missing, incomplete, or hard to locate". In paticular, I will give two proofs of the following "folk theorem": if $alpha$ is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear fractional transformation of $alpha$ also has bounded partial quotients. The first proof is based upon arithmetics arguments and the second one upon the geometrical interpretation of the best approximations to $alpha$. The result is a consequence of the following inequality due to Lagarias and Shallit [#!LS!#]: begin{displaymath}frac{1}{vert ad-bcvert}L(alpha) leq Lleft(frac{aalpha+b}{calpha+d} ight)leqvert ad-bcvert L(alpha),,end{displaymath} where $a, b,c, d ; in {mathbb Z},$ with $vert ad-bcvert ot=0$ and $L(alpha)$ is the Lagrange constant of $alpha$.