Let (Sn) be a nondecreasing sequence of positive real numbers and define Tn= (1/n)(s1+s2+· · ·+sn)for n ∈ N. Prove that (Tn) is a nondecreasing sequence.
Sorry, I am not good with Latex....so here is the proof informally.
Given that Sn is a non-decreasing sequence of positive numbers, each subsequent term is greater than the previous for all 'n' in N.
Then I wrote down several terms for the sequence Tn:
T1 = (s1)/1
T2 = (s1+s2)/2
T3 = (s1+s2+s3)/3
and noticed that T2 - T1 was positive , T3 - T2 was positive and so on...
Since we are proving that Tn is a non-decreasing sequence as well...its every subsequent term has to be greater than the previous.
T(n+1)-Tn = [s1+s2+...+sn+s(n+1)]/(n+1) - [s1+s2+s3+....+sn]/n
Now to subtract make the common denominator, and get
[n*Sn+1 -(s1+s2+s3...+sn)]/ (n*(n+1))
This is greater than or equal to zero because we have n*Sn+1 and each is greater than each s1 thru. sn term.
Then I used induction to show that
T(n+1)+1 - Tn+1 is also greater than or equal to zero. Therefore the sequence must be non-decreasing.
Hope this is understandable.