We know that a function $f: X \to Y$, where $X$ and $Y$ are metric spaces with metrics $d_1$ and $d_2$ respectively, is a Lipschitz function if there exists a real number $K>0$ such that for every $x,y$ that belong to $X$,
$$ d_2( f(x), f(y)) \le K d_1(x, y).$$
Therefore, the negation should look as follows:
A function $f: X \to Y$ is not a Lipschitz function if for every real number $K>0$ there exists some $x,y$ that belong to $X$ such that
$$ d_2( f(x), f(y)) > K d_1(x, y).$$
But now I want to write an equivalent negation using sequences, how can I do that?
$f$ is not Lipschitz if and only if for every positive integer $n$ there exist $x_n,y_n$ such that $$d_2(f(x_n,f(y_n)) >n d_1(x_n,y_n).$$