It is easier to see this if you change C and k separately. So, suppose that

|f(x)| <= C|g(x)| for all x > k

Also, suppose that C' >= C and k' >= k. Since |f(x)| <= C|g(x)| implies |f(x)| <= C'|g(x)| for any x, we have that

|f(x)| <= C'|g(x)| for all x > k

Now if some property holds for all x > k, and if k' >= k, then this property holds for all x > k'. This is because the set {x | x>= k'} is a subset of {x | x>= k}.