An inductive limit of amenable groups is amenable

Question

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one.

Thanks in advance.

Edit: I wanted it for a directed Limit over a directed poset, not just a totally ordered one.

Answer 1

This is Theorem 3.4 in A Survey of Amenability Theory for Direct-Limit Groups. If you are interested in arbitrary colimits, see also Theorem 3.3 (3).