We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove that two irrational numbers $\alpha$ and $\beta$ are congruent if and only if the tails of their infinite continued franctions expansions eventually agree, this means: $$\alpha=[\alpha_0;\alpha_1,\alpha_2,...,\alpha_n,e_1,e_2,...]$$ and $$\beta=[\beta_0;\beta_1,\beta_2,...,\beta_j,e_1,e_2,...]$$
So, I started doing the second direction (If their continued fractions eventually coincide, then they are congruent), but I got to here and I dont know what else to do: $$\alpha=(\alpha_0+\alpha_1+...+\alpha_n)+\beta+(\beta_0+\beta_1+...+\beta_j)$$ (I know all of the $\alpha_i$ and $\beta_k$ are integer). I need help in both directions, and I dont know where to start the other one.