Thread: Proof By Induction

I know induction is suppose to be easy, but this one is stumping me because it is about inequalities.

let s_n be a positive non decreasing sequence, i.e s_n < s_n+1 for all n

let q_n be the sequence 1/n *(s_1 + s_2 + ...+ s_n)

prove that the sequence q_n is also non decreasing , ie q_n < q_n+1 for all n

I was able to do the base case (obviously), and then i said assume q_n < q_n+1 for some n. and i wrote a lot of scratch but cant seem to prove q_n+1 < q_n+2 (the induction hypotheses)

Okay, this is sort of crude, but I think it works.













where the fourth line follows by properties of arithmetic mean (the average of a set of values is less than or equal to the maximum of the values) (and hence ).

EDIT: I'm not sure if this is technically an inductive proof, since it makes no use of the assumption .