Prove language is not regular? I know how to use the pumping lemma to do so, but I don't think that can be used for this language:

$$L = \{x \in \{0,1\}^* : \text{no prefix of $x$ has more $1$'s than $0$'s}\}. $$

What other method can I use to prove this?
Hint: use $0^n1^n$ for large enough $n$.

Another approach is to use the Myhill-Nerode criterion, say with the words $0^n$.