About a pre-processing step for primal-dual weighted set cover problem

Question

I was reading the paper titled "Primal-dual RNC approximation algorithms" by Rajagopalan and Vazirani. I have a problem of understanding the Lemma 4.1.1.

They present a dual fitting based algorithm for weighted set cover. First let me set up the required concept to clarify where I am having trouble. Suppose we have $n$ elements ($U$) and $m$ sets ($S$). Each set has a positive weight. Let $E_v$ holds the sets in which the element $v$ is present. Let $\beta =$max$_{v \in U}$ min$_{s \in E_v}$ weight(s). Let also $IP^*$ is the weight of an optimal set cover. It is easy to see, $IP^*\geq \beta$.

Now assume we have an approximation algorithm for weighted set cover. What the paper is saying in the lemma is that you can do a pre-processing before starting the approximation algorithm as follows. You can scan through the sets and add any sets that have weight $\leq \beta/n$. Since there are $n$ elements the additional cost is at most $\beta$. Then they claim that

Since $\beta$ is a lower bound on $IP^*$, this cost is subsumed in the approximation. And this is the statement I did not understand.