In the book about K-theory (a friendly approach) by Wegge and 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?

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?