Please take a look at the proof I added, there are some things I do not understand with this proof. 1. Does it really show that |f(x)-f(y)|≤d(x,y) for all x and y? Or does it only show that if there is an ball with radius r around x, and this ball is contained in an O in the open covering, and if y is contained in that ball. Then |f(x)-f(y)|≤d(x,y)? 2. Also please take a look at the sentence I underlined with a red line. Why can he say that f(y)≥f(x)-d(x,y)? I could understand if he wrote r instead of f(x), that is f(y)≥r-d(x,y), why is he allowed to use r instead of f(x)?