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Let (an)n∈N be any sequence of real numbers. Which of the following statements are true?

Give precise references to the results in the Lecture Notes for those which are true. Construct counter examples for those that are false.

(i) Every sequence (an)n∈N has a convergent subsequence.

(ii) If (an)n∈N has a convergent subsequence, then (an)n∈N is convergent.

(iii) If (an)n∈N is bounded, then (an)n∈N has a convergent subsequence.

(iv) If (an)n∈N has a convergent subsequence, then (an)n∈N is bounded.

My answers:

i) False: {1,2,3,4,5,6...) has no convergent subsequence.

ii) False: {-1/2, 1/4, -1/8, 1/16, -1/32....} diverges but has subsequence {1/4, 1/16, 1/64...} which converges

iii) True (Bolzano Weierstrass theorem)

iv) False: {1, 1, 2, 1/2, 3, 1/3....} is unbounded but has subsequence {1, 1/2, 1/3, 1/4...} which converges