  Classical result states that for two continuous and strict means $M,\,N
\colon I^2 \to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant
mean $K \colon I^2 \to I$, i.e. such a mean that $K \circ (M,N)=K$ and,
moreover, the sequence of iterates $((M,N)^n)_{n=1}^\infty$ converge to $(K,K)$
pointwise.
  Recently it was proved that continuity assumption cannot be omitted in
general. We show that if $K$ is a unique $(M,N)$-invariant mean then, under no
continuity assumption, $(M,N)^n \to (K,K)$.
