Anexample of a sequence with the given property is provided inred.... I need hel
ID: 2937727 • Letter: A
Question
Anexample of a sequence with the given property is provided inred.... I need help proving that they really do have the requiredproperties. a)Asequence that has subsequences that converge to 1,2, and3. {1,2,3,1,2,3,1,2,3,...} b)Asequence that has subsequences that converge to and-. {1,-1,2,-2,3,-3,4,-4,5,-5,...} c)Asequence that has a strictly increasing subsequence, a strictlydecreasing subsequence, anda constant subsequence. {0,1,-1,0,2,-2,0,3,-3,0,4,-4,...} d)Anunbounded sequence which has a convergentsubsequence. {0,1,0,2,0,3,0,4,0,5,0,6,...} e)Asequence that has no convergentsubsequence. {0,1,2,3,4,...}Note: (b) can be proven with out piece wise, (c) and (d) should beproven piece wise, (e) I have no idea
Anexample of a sequence with the given property is provided inred.... I need help proving that they really do have the requiredproperties. a)Asequence that has subsequences that converge to 1,2, and3. {1,2,3,1,2,3,1,2,3,...} b)Asequence that has subsequences that converge to and-. {1,-1,2,-2,3,-3,4,-4,5,-5,...} c)Asequence that has a strictly increasing subsequence, a strictlydecreasing subsequence, anda constant subsequence. {0,1,-1,0,2,-2,0,3,-3,0,4,-4,...} d)Anunbounded sequence which has a convergentsubsequence. {0,1,0,2,0,3,0,4,0,5,0,6,...} e)Asequence that has no convergentsubsequence. {0,1,2,3,4,...}
Note: (b) can be proven with out piece wise, (c) and (d) should beproven piece wise, (e) I have no idea
Explanation / Answer
QuestionDetails: Anexample of a sequence with the given property is provided inred.... I need help proving that they really do have the requiredproperties. a)Asequence that has subsequences that converge to 1,2, and3. {1,2,3,1,2,3,1,2,3,...}HERE TAKE 3 SUBSEQUENCES AS GIVEN BELOW
1.SUBSEQUENCE OF EVERY III TERM STARTING FROM I TERM. WE GET
[1,1,1,1...........] WHICH CONVERGES TO 1
2.SUBSEQUENCE OF EVERY III TERM STARTING FROM II TERM. WEGET
[2,2,2,2...........] WHICH CONVERGES TO 2
3.SUBSEQUENCE OF EVERY III TERM STARTING FROM III TERM. WEGET
[3,3,3,3...........] WHICH CONVERGES TO 3
b)Asequence that has subsequences that converge to and-. {1,-1,2,-2,3,-3,4,-4,5,-5,...}
1.SUBSEQUENCE OF EVERY II TERM STARTING FROM I TERM. WEGET
[1,2,3,4...........] WHICH CONVERGES TO
2.SUBSEQUENCE OF EVERY II TERM STARTING FROM II TERM.WE GET
[-1,-2,-3,-4...........] WHICH CONVERGES TO - c)Asequence that has a strictly increasing subsequence, a strictlydecreasing subsequence, anda constant subsequence. {0,1,-1,0,2,-2,0,3,-3,0,4,-4,...}
1.SUBSEQUENCE OF EVERY III TERM STARTING FROM II TERM. WEGET
[1,2,3,4...........] WHICH IS A a strictly increasingsubsequence
2.SUBSEQUENCE OF EVERY III TERM STARTING FROM III TERM.WE GET
[-1,-2,-3,-4...........] WHICH IS A a strictly DECREASINGsubsequence
1.SUBSEQUENCE OF EVERY III TERM STARTING FROM I TERM.WE GET
[0,0,0,0...........] WHICH IS A a CONSTANTsubsequence
d)Anunbounded sequence which has a convergentsubsequence. {0,1,0,2,0,3,0,4,0,5,0,6,...}
1.SUBSEQUENCE OF EVERY II TERM STARTING FROM I TERM. WEGET
[0,0,0,0...........] WHICH IS A convergentsubsequence
e)Asequence that has no convergentsubsequence. {0,1,2,3,4,...}
1.SUBSEQUENCE OF EVERY II TERM STARTING FROM I TERM. WEGET
[0,2,4,6...........] WHICH IS A DIVERGENTsubsequence
SIMILARLY WE CAN SEE THAT ANY SUBSEQUENCE OF III TERMS OR WHATEVERNATURE
IS A DIVERGENT SBSEQUENCE