In the book _K-Theory and $C^*$-Algebras: A Friendly Approach_ by Niels Wegge-Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(a_n)_n$ so that $a_{n+1}a_n=a_n$". In the next lemma the only assumption about $A$ is that $A$ is $\sigma$-unital, which means that there is some countable approximate unit. However the authors claim that in this case we may assume that this approximate unit is as above. 
>Why we can assume that this approximate unit have this additional property (having assumed that $A$ is just $\sigma$-unital?