Theorem 04 Every unbounded sequence has a monotone subsequence that diverges

# Theorem 04 every unbounded sequence has a monotone

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Theorem 0.4 Every unbounded sequence has a monotone subsequence that diverges either to + or to -∞ . 80
Proof: Assume that ( s n ) is unbounded above (for the case of bounded below, we can use the similar argument). We can find s n 1 from ( s n ) such that s n 1 1 . Secondly we can find s n 2 from ( s n ) such that s n 2 2 , with n 2 > n 1 . Next we can find s n 3 from ( s n ) such that s n 3 3 , with n 3 > n 2 . Repeating this process, we obtain a subsequence s n k + . Limit Superior and Limit Inferior Recall that we have notions of sup S and inf S for any subset S R . We want to consider the case where S is a sequence. Definition Let ( s n ) be a bounded sequence. A number α is a subsequential limit of ( s n ) if there is a subsequence ( s n k ) of ( s n ) such that s n k α [Examples and remarks] 1. Let s n = 1+( - 1) n +1 2 and ( s n ) = (1 , 0 , 1 , 0 , 1 , 0 , ... ). Then 0 and 1 are subsequencial limits. 2. Let s n = ( - 1) n n +1 n and ( s n ) = ( - 1 2 , 3 2 , - 4 3 , 5 4 , - 6 5 , ... ). Then - 1 and 1 are sunsequencial limits. 3. Let s n = - 2 + 1 n , n = 3 k - 2 , 1 n , n = 3 k - 1 , 1 + 1 n , n = 3 k. Then - 2 , 0 and 1 are subsequenctial limits. 4. Let s n = sin 6 , and ( s n ) = 1 2 , 3 2 , 1 , 3 2 , 1 2 , 0 , - 1 2 , - 3 2 , - 1 , - 3 2 , - 1 2 , 1 2 , ... . Then 1 2 , 3 2 , 1 , 0 , - 1 2 , - 3 2 , - 1 . are subsequenctial limits. 81
Let ( s n ) be a bounded sequence. Let S = { α | α is a subsequential limit of ( s n ) } . Then 1. S 6 = . 2. S is a bounded set. By Theorem 0.4, every bounded sequence has a convergent subsequence. Then S is not an empty set. Since ( s n ) is bounded, so is S . Definition Let ( s n ) be a bounded sequence and let S be the set of its all subsequential limits. 1. The limit superior of ( s n ) (denoted by lim sup s n ) is lim sup s n = sup S. 2. The limit inferior of ( s n ) (denoted by lim inf s n ) is lim inf s n = inf S. [Examples and remarks] 1. In some books, we use the notation: lim = lim sup and lim = lim inf. 2. lim inf s n lim sup s n . 3. If ( s n ) is convergent (i.e., s n s ), since every its subsequence converges to the same limit s , we see S = { s } . In this case, lim inf s n = lim sup s n . if ( s n ) is not convergent, then we must have lim inf s n < lim sup s n . In this case, we say that ( s n ) oscillates .

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• Fall '08
• Staff
• Integers, Limit of a sequence, subsequence, Snk, lim snkm