Let $ {Xn}$ be a sequence of compact length spaces, $GHlim X_n=X$ (Gromov-Hausdorff convergence), and let $f_n$ be as in Exercise 7.5.9 (Burago and Burago and Ivanov). Prove that, for all large enough n, $f_n$ induces a surjective homomorphism of fundamental groups.