Let sn be an increasing sequence of positive numbers and defineOn = 1nS1 + S2 + + Sn Prove On is an increasing sequence

To prove that (On) is an increasing sequence, we need to show that On+1 ≥ On for all n.

Let's start by expanding the expression for On+1:

On+1 = 1/(n+1)(S1 + S2 + ... + Sn + Sn+1)

We can rewrite this as:

On+1 = (n/n+1)On + Sn+1/(n+1)

Now, we need to show that On+1 ≥ On. We can do this by showing that the second term on the right-hand side is greater than or equal to zero:

Sn+1/(n+1) ≥ 0

This is true because Sn+1 and n+1 are both positive.

Therefore, we just need to show that the first term on the right-hand side is greater than or equal to On:

(n/n+1)On ≥ On

Simplifying this inequality, we get:

On ≤ Sn+1/(n+1)

This is true because (sn) is an increasing sequence, so Sn+1 ≥ Sn. Therefore, Sn+1/(n+1) ≥ Sn/n, and since (sn) is increasing, Sn/n ≥ On.

So we have shown that both terms on the right-hand side of the inequality On+1 ≥ On are greater than or equal to On, and therefore On+1 ≥ On for all n. This proves that (On) is an increasing sequence