Question: a) Let {x n } be a bounded sequence. Prove that if limn sup |x n | = 0, then limn x n exists and

a) Let {xn} be a bounded sequence. Prove that if limn sup |xn| = 0, then limn xn exists and equals 0.

(b) Prove that a bounded sequence that does not converge always has at least two subsequences that converge to different limits.

  1. (c) Find the limit inferior and limit superior of the sequence {an} if an = sin n for all n N.
  2. (d) Find the set of all subsequential limits for the sequence {xn} if for all n N

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