Have I done this right? I have shown that every element of Σ exists in Σ*, so is it ok to do what I did in step 5?
If Σ = {s,i, n, g}, show that singing is in Σ*.
Solution: (in 5 steps)
1) Since λ ∊ Σ* and i ∊ Σ, i ∊ Σ*.
2) Since i ∊ Σ* and n ∊ Σ, in ∊ Σ*.
3) Since in ∊ Σ* and g ∊ Σ, ing ∊ Σ*.
4) Since ing ∊ Σ* and s ∊ Σ, sing ∊ Σ*.
5) Since sing ∊ Σ*, and ing ∊ Σ*, singing ∊ Σ*.
This can't be optimal. Either your definition of the Kleene star permits arbitrary concatenation, in which case you can do everything in one step, or it doesn't, in which case you need seven steps. Your step 5 is essentially using the theorem that $\Sigma^\star=\Sigma^\star \Sigma^\star$.