A Result on Convergence of Sequences of Iterations with Applications to Best-Response Dynamics

The result that says the sequence of iterations $x^{k+1}=f(x^k)$ converges if $f:[0,1]\to [0,1]$ is an increasing function has numerous applications in elementary economic analysis. I generalize this simple result to some mappings $f:S\subset {[0,1]}^n\to S$. The applications of the new result include the convergence of the best-response dynamics in the general version of the Crawford and Sobel model and in some versions of the Hotelling and Tiebout models.