Equivalences for a Banach lattice

Question

I'm trying to prove the following equivalences for a Banach lattice $E$:

1. $E$ has an order continuous norm
2. Every monotone order bounded sequence in $E$ is convergent
3. E is an ideal in $E^{**}$

I have been able to prove that $1) \implies 2)$ but I wasn't able of doing much more. I was trying to prove that $2) \implies 3)$ and that $3) \implies 1)$ but I got nothing.

What is the best way to prove the above equivalences? Any hint would be greatly appreciated.

I have found in Lindenstrauss' Classical Banach Spaces that the third statement is a Theorem whose proof is bases in Theorem 1.b.14 of that book that states an isometric equivalence between order continuous lattices with a weak unit and a particular kind of function spaces, but I would like to be able to prove the above statement without needing to use that theorem.