Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself? 
Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself?

This is the converse of if the initial ideal of an ideal is radical, then the ideal is radical.
Also, how can I create more examples?

Let $R=K[X,Y,Z]$ and $I=(X^2-YZ)$. If one considers the lexicographical order on $R$ with $X>Y>Z$ then $\operatorname{in}(I)=(X^2)$.
