(a) ｢b⸤i⸥ Σi=1,∞｣ converges
⇒ ｢a⸤i⸥ Σi=1,∞｣ converges
(b) ｢a⸤i⸥ Σi=1,∞｣ diverges
⇒ ｢b⸤i⸥ Σi=1,∞｣ divergesᐥ
Proof of (a)
┆0 ≤ a⸤n⸥┆
｢a⸤i⸥ Σi=1,n｣ is increasing.
｢a⸤i⸥ Σi=1,n｣
┆a⸤n⸥ ≤ b⸤n⸥┆
≤ ｢b⸤i⸥ Σi=1,n｣
┆0 ≤ b⸤n⸥┆
≤ ｢b⸤i⸥ Σi=1,∞｣
Thus ｢a⸤i⸥ Σi=1,n｣ is bounded above.
┄
A bounded monotone sequence is convergent.
┄
So ｢a⸤i⸥ Σi=1,n｣ converges.
(b) is the contraposituve of (a).
