Let $A$ be a $C^*$-algebra and $a,b \in A$ positive elements. We define a relation (Cuntz sub-equivalence) by saying $$a\lesssim b: \Leftrightarrow \exists\, (r_n)_{n\in\mathbb{N}}\subset{A}\text{ such that } \|r_n^* b r_n-a\|\rightarrow 0. $$ I want to show, that this relation is transitive. The only proof I can come up with requires the sequence $(r_n)_n$ to be bounded.

Thank you