Please answer it clearly We say a sequence is monotonic if i

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|

Solution

Because,

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

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