# approximation – Trouble to understand the proof of greedy algorithm for set cover

Problem definition: Given a universe $$U$$ of $$n$$ elements, a collection of subsets of $$U$$, $$S = {S_1,…, S_k}$$, and a cost function $$c: S to Q^{+}$$. Find a minimum cost subcollection of $$S$$ that covers all elements of $$U$$.

The provided algorithm (Approximation algorithms – Vijay V. Vazirani)

Part of the proof where I have trouble to understand

My question

I have a difficult time to understand the last in equality, if $$|bar{C}| leq n – k + 1$$, why does $$cfrac{OPT}{|bar{C}|} leq cfrac{OPT}{n – k + 1}$$ hold?