Decomposing sequences into monotonic subsequences

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically increasing if and only if every set of k+1 positive integers contains two integers x,x' with x < x' such that f(x) <= f(x'). The function f:X \to Y is called k-monotonic if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing or monotonically decreasing for i=1,...,k. It is also proved that there does not exist a k-monotonic function from N onto Q.