In Theorem 2.41, the one with the heine borel theorem, I do not understand the second inequality, |x_n - y| >= |x_0 - y| - 1/n >= 1/2*|x_0 - y| for all but a finite n. why did the 1/n turn |x_0 - y| into 1/2*|x_0 - y|? I understand this inequality to be using the triangle inequality to show that if the set S has x_0 as a limit point, then by its definition it cannot contain any other limit points, but I don't get how this inequality was made. If this is true for all but a finite n, then it means that once n becomes big enough, then this inequality is true right? When is n big enough? Why does the inequality not reduce to just |x_0 - y|, as once n becomes big enough 1/n is just 0.