The extreme points of a convex hull of any set is the same with the extreme points of this set?

I consider any set $$A$$ in $$\mathbb{R}^n$$. Note that $$A$$ is not necessary to be a convex set. I define the extreme point as follows: Given any set $$C$$, a point $$a\in C$$ is an extreme point of the set $$C$$ if we cannot find any two different points $$a_0\in C,a_1\in C$$ and $$a_0\neq a_1$$ such that $$a=\lambda a_0 + (1-\lambda) a_1$$, for any $$\lambda\in(0,1)$$

My question is: whether or not, the set of the extreme points of $$A$$ is the same as the set of the extreme points of $$conv(A)$$, where $$conv(A)$$ means the convex hull of set $$A$$.

I guess the two sets of extreme points above would be the same set since the operator $$conv()$$ should not generate or remove any extreme points of $$A$$. If the two sets are different, that means there exists a point in $$conv(A)$$ such that the point is the extreme point of $$conv(A)$$ but not the extreme point of $$A$$. I think this leads to a contradiction but I cannot vigorously argue it.