The socalled $K\epsilon$ principle is as follows:
Let $(a_n)$ be a sequence, and let $L$ be some number. Assume for every $\epsilon>0$ there exists $N\in\mathbb{N}$ such that $|a_n-L|<K\epsilon$ for all $n\geq N$, where $K$ is a positive real number that does not depend on $\epsilon$ or $N$. Then $(a_n)$ converges to $L$.
If I write the assumption in logic notation, should it be $$ \exists K>0\forall \epsilon>0\exists N\in \mathbb{N}(n\geq N\implies |a_n-L|<K\epsilon)? \tag{1} $$ I am not sure, if there should be $\forall K>0$ or $\exists K>0$. On the other hand, if $(a_n)$ converges, say, to $M$, then $(1)$ holds, where $L$ is replaced by $M$. For example, we can make $|a_n-M|$ to be even more smaller than $\epsilon$, say $\epsilon/2$ by choosing $n$ sufficiently large. In this case, we have $K=1/2$. Is my understanding correct in both directions?