Let sn be an increasing sequence of positive numbers and defineOn = = S1 + S2 + + Sn Prove On is an increasing sequence

To prove that (On) is an increasing sequence, we need to show that On+1 is always greater than or equal to On.

We can write On+1 as:

On+1 = S1 + S2 + ... + Sn+1

Since (sn) is an increasing sequence, we have:

S1 ≤ S2 ≤ ... ≤ Sn ≤ Sn+1

Therefore,

On+1 = S1 + S2 + ... + Sn + Sn+1 ≥ S1 + S2 + ... + Sn = On

Hence, we have shown that On+1 is greater than or equal to On, which implies that (On) is an increasing sequence.