# Abelian/Finite Projections in Von Neumann Algebras

I know that provided a von Neumann Algebra acting on Hilbert $$(\mathcal{M},\mathcal{H})$$ a projection $$e \in P(\mathcal{M})$$ is said to be abelian if $$e\mathcal{M} e$$ is $$\textbf{abelian}$$ and the same is said to be $$\textbf{finite}$$ if $$f \le e$$ and $$f \sim e \Rightarrow f = e$$ and I should prove the following statements:

1)Any abelian projection is finite

2)if $$e$$ is abelian and $$f \preceq e$$ then f is abelian ($$\preceq$$ means that $$f$$ is equivalent to a subprojection of e)

3)if $$e$$ finite and $$f \preceq e$$ then $$f$$ is finite

Can you give me any hints? For the first one I was thinking that if $$e$$ is abelian then $$e\mathcal{M} e = \mathcal{Z}(\mathcal{M})$$...