Establish which of the following statements are true. Prove or give a counter ex
ID: 3110956 • Letter: E
Question
Establish which of the following statements are true. Prove or give a counter exampleA. A sequence is convergent if and only if all of its subsequences are cnvergent.
B. A squuence is bounded if and only if all of its subsequences are bounded.
C. A sequence is monotonic if and ony if all of its subsequences are monotonic.
D. A sequence is divergent if and only if all of its subsequences are divergent.
Establish which of the following statements are true. Prove or give a counter example
A. A sequence is convergent if and only if all of its subsequences are cnvergent.
B. A squuence is bounded if and only if all of its subsequences are bounded.
C. A sequence is monotonic if and ony if all of its subsequences are monotonic.
D. A sequence is divergent if and only if all of its subsequences are divergent.
Explanation / Answer
A) The statement is True.
Let an converges to a, and let ajn be a subsequence of an. For any > 0, there is N such that if n N, then |an a| < . We have that jn n for all n 1. Thus, if n N, then jn n N and |ajn a| < . Therefore, ajn converges to a. Thus, any subsequence of an converges to a.
Converse: Suppose, all subsequences of an converge to a. But an is a subsequence of itself and thus an converges to a. Therefore, an converges to a.
B) The statetment is true.
If |an |<B, n. then |ank | <B, nk. If |an| is eventually larger than any P> 0, then |ank | is eventually larger than P.
C) The statetment is true.
If a sequence is monotone, then the subsequence maintains the order of its terms,and the subsequence isalso monotone. If a sequence contains a subsequence that is not monotone, then the original sequence is not monotone since the order of terms is preserved.
D) The statement is not true.
For a convergent sequence, every subsequence converges to the same limit. But, if we have at least one divergent subsequence, then sequence cannot converge. So, it is not required for all the subsequences to be divergent for the divergent of the sequence.
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