in my "discrete mathematics and its applications" book it is written:

I dont see why the "proof" is true: if we choose a k' > k and k' is really A LOT bigger than k and if we choose at the same time a C' which is just a little bit bigger than C - why are we still "winning" ?|f(x)| <= C|g(x)| whenever x > k

"f is big oh of g"

the constants C and k in the definition of bigO notation are called witnesses to the relationship f is O(g). To establish that f is O(g)we need only one pair of witnesses to this relationship. ... Note that when there are one pair of witnesses to the relationship f is O(g), there areinfinitely many pairs of witnesses.To see this, note that if C and k are one pair of witnesses, then any pair C' and k' where C < C' and k < k', is also a pair of witnesses, since

|f(x)| <= C|g(x)| = C'g(x) whenever x > k' > k.