Preuve simplifi\'ee du th\'eor\`eme de Serret sur les nombres \'equivalents

If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{Z})$. The converse is Serret’s well-known theorem, but we give a simpler and purely algebraic proof of it.