I do not know why do I need a claim 1 here, I mean I cannot see any relationship between claim 1 and the question: prove T(n) = O (n). And in the end the result is C>=20, why does this prove T(n) = O (n)?

Problem OLD.1.1: Consider the recurrence S T([n/3]) +T([n/4]) +T([n/5]) +T([n/6]) +n 11 n > 6 n 20, and for all n > 1, we have T(n) 1 by definition. Induction hypothesis. Let n > 6. Assume that T(k)