We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2(n+2)/4 best possible. Given a real vector x =(x 1,..., x n−1, 1) εℝn this CFA generates a sequence of vectors (p 1 (k))..., p n−1 (k), q k) εℤn, k = 1, 2,... with increasing integers ¦q (k)¦ satisfying for i = 1,..., n − 1 $$\left| {x_i - p_i ^{(k)} /q^{(k)} } \right| \leqslant 2^{(n + 2)/4} \sqrt {1 + x_i^2 } /\left| {q^{(k)} } \right|^{1 + \tfrac{1}{{n - 1}}} .$$ By a theorem of Dirichlet this bound is best possible in that the exponent $$1 + \tfrac{1}{{n - 1}}$$ can in general not be increased.