MHB Confirm Answers on Homework Sheet: Subsequence Convergence

[Carla1985]

Question from my homework sheet. Can someone confirm I've got these correct.

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

 

[Ackbach]

Looks good to me.

[EDIT] See Evgeny.Makarov's post below for a correction.

 

[Evgeny.Makarov]

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

Does -1/2, 1/4, -1/8, 1/16, -1/32, ... really diverge?

 

[Ackbach]

Does -1/2, 1/4, -1/8, 1/16, -1/32, ... really diverge?

So maybe try flipping what the alternating terms are doing:
{-1/2, 1/2, -1/4, 1, -1/8, 2, -1/16, 4, ...}