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?

strictly positiveelement: I have problems in understanding what does it mean in the nonunital case (in the unital my guess would be positive and invertible) and how to obtain such an element (my guess is that here the assumption of being $\sigma$-unital is used and one has to take something like $\sum_{n=1}^{\infty}\frac{1}{2^n}e_n$ where $(e_n)_n$ is some countable approximate unit). $\endgroup$ – truebaran Jun 26 '16 at 18:56