Closure properties of E-local spectra

Exercise 7.1.4. Show the following basic properties of \(E\)-local spectra:

(1)

A spectrum \(L\) is \(E\)-local if and only if \(\hom (X,L) = 0\) for every \(E\)-acyclic spectrum \(X\);

(2)

Given an exact sequence \(X \to Y \to Z\), if two of these three spectra are \(E\)-local then so is the third;

(3)

In particular, if \(L\) is \(E\)-local then so are all its shifts \(L[n]\);

(4)

A limit \(\lim _i L_i\) of \(E\)-local spectra is \(E\)-local;

(5)

A retract of an \(E\)-local spectrum is \(E\)-local.