regular languages and suffixes

Question

Let $\Sigma$ be a finite alphabet. For a language $L \subseteq\Sigma^*$ we define: $$Suff(L)= \left\{x\in\Sigma^*| \exists u \in \Sigma^*, u\cdot x \in L\right\}$$ Show an example of a language $L$ that is not regular for which $Suff(L)$ is regular.
I took $L=\left\{0^n1^n | n\geq 0 \right\}$.
It's easy to show that $L$ is not regular using the pumping lemma, however I tried to show that $Suff(L)$ is regular with no success.
I tried to split the possibilities of suffixes of every word $w \in L$ by cases.
If anyone have any idea how to show it, it would be great (meaybe I'm even wrong and my example is not true and that's why I can't prove it)