Please answer it clearly. We say a sequence is monotonic if it is either increas

Question

Please answer it clearly. We say a sequence is monotonic if it is either increasing or decreasing. A sequence is bounded if M R such that |a_n| lessthanorequalto M, n N. Monotonic Sequence Theorem. All sequences which are bounded and monotonic converge. Proof. Without loss of generality, let {a_n} be an increasing and bounded sequence and let L = sup_n N a_n. Fix > 0. Then, by the definition of the supremum (least upper bound), N N such that a_N > L - As {a_n} is increasing and bounded above by L, we have L greaterthanorequalto a_n greaterthanorequalto a_N, n greaterthanorequalto N. Thus |a_n - L| lessthanorequalto |a_N - L|

Explanation / Answer

Because,

If {bn}is a decreasing bounded sequence then the sequence {-bn} is an increaing sequence and by the above theorm it converges. If  {-bn} converges to K then  the original sequence {bn} converges to -K.

Therefore it is sufficient to prove the above theorem for any one case.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.