Sometimes true Let s n N 1 2 3 4 Every subsequence of s n is unbounded Let s n

# Sometimes true let s n n 1 2 3 4 every subsequence of

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Sometimes true: Let s n = N = (1 , 2 , 3 , 4 , · · · ). Every subsequence of ( s n ) is unbounded. Let s n = 1 , n odd 1 /n, n even . Then ( s 2 n 1 ) = (1 , 3 , 5 , 7 , · · · ) is un- bounded; ( s 2 n ) = (1 / 2 , 1 / 4 , 1 / 6 , 1 / 8 , · · · is bounded (and has limit 0). (e) If ( s n ) is a bounded, monotone sequence, then ( s n ) is a Cauchy sequence. Always true: A bounded monotone sequence is convergent; a convergent sequece is a Cauchy sequence. 2
(f) If ( s n ) is a bounded sequence, then ( s n ) has a Cauchy subsequence. Always true: This is Theorem 2, Section 19. (g) Every oscillating sequence has a convergent subsequence. Always true: If ( s n ) oscillates, then α = lim sup s n ̸ = lim inf s n = β , and there is a subsequence s n k which converges to α and a subsequence s n p which converges to β , α ̸ = β . (h) Every oscillating sequence diverges. Always true: There are subsequences which converge to two di ff erent limits, as shown above. 3
(i) If ( s n ) is an unbounded sequence, then either s n + or s n → −∞ . Sometimes true: s n = N is an unbounded sequence and s n + . s n = n, n odd 1 /n, n even is unbounded, but s n does not diverge to + or to −∞ . (j) If ( s n ) is a bounded sequence and α = sup { s n } , then ( s n ) has a subsequence which converges to α . Sometimes true: If ( s n ) is bounded and increasing, then s n α = sup { s n } Let s n = (2 , 0 , 1 / 2 , 2 / 3 , 3 / 4 , · · · , n 1 n , · · · ). Then sup { s n } = 2, ( s n ) does not have a subsequence which converges to 2 (all subsequences converge to 1).

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• Fall '08
• Staff
• Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, Sn, subsequence
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