I want to show that $K_0(A)$ is a countable abelian group when $A$ is a separable $C^*$-algebra and I know that this is the case when $A$ is a unital separable $C^*$-algebra as this has been shown earlier where the proof I have seen is similar to this one: Show that $K_0(A)$ is a countable group if $A$ is a unital, separable C* algebra. However, it is not clear to me where in the linked proof one uses that $A$ is unital, so my question is: How different is the proof when $A$ is not necessarily unital?
I also know that $K_{00}(A)$ is a countable abelian group when $A$ is any separable $C^*$-algebra so I guess this could be used somehow but I am not sure how to or if I can use this fact to something.
Show that if $A$ is separable, then so is its unitization $\tilde A$. Then $K_0(\tilde A)$ is countable, and since the map $K_0(A)\to K_0(\tilde A)$ is injective, the result follows.