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Which of the following statements are true:
  • a)
    0 Every bounded sequence is convergent
  • b)
    Every convergent sequence is bounded
  • c)
    If an →a , then an2 →a2 as n→∞
  • d)
     
    If an2 →0, then an →o as n→∞
Correct answer is option 'C,D'. Can you explain this answer?
Most Upvoted Answer
Which of the following statements are true:a)0 Every bounded sequence ...
Note that every convergent sequence is bounded, But converse need not be true. as let an = (-1)n be a bounded sequence, But not convergent.
.:. option (A) is incorrect and option (B) is correct.
Note that if xn .and yn are two convergent sequences. such that lim xn = x and lim yn =y.
Then by algebra of of convergent sequence  is also convergent and lim  = lim xn lim yn = xy
Here let xn = yn = an ∀n then x = y = a , and hence lim an2 = lim an . an = lim an.liman = a·a = a2
Hence option (c) is correct.
For option (d), since , then by definition for given ∈> 0 there exist a natural number K such that if , then 0 
Since ∈ > 0 is arbitary, this implies lim an = 0
Hence, option (d) is correct.
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Which of the following statements are true:a)0 Every bounded sequence is convergentb)Every convergent sequence is boundedc)If an→a , then an2→a2as n→∞d)If an2→0, then an→o as n→∞Correct answer is option 'C,D'. Can you explain this answer?
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