TITLE: Approximation Property
QUESTION [5 upvotes]: It seems to be a folk result that l_infinity has the approximation property, even the bounded approximation property, and also, I think, even the so-called propery pi (approximation property) of Lindenstrauss. 
This is alluded to in a few texts, but I cannot seem to find the proof, which is presumably obvious. 
Does anyone have a reference or an easy solution?
Cheers,
R. Fry

REPLY [11 votes]: Partition the measure space into finitely many sets and consider the span of their indicator functions.  This space is isometrically isomorphic to $\ell_\infty^n$ for appropriate $n$ and hence is the range of a norm one projection.  Index the partitions by refinement to get a net of norm one finite rank projections that converge strongly to the identity.