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Carla1985

Member
Feb 14, 2013
93
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

Indicium Physicus
Staff member
Jan 26, 2012
4,192
Looks good to me.

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

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492

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

Indicium Physicus
Staff member
Jan 26, 2012
4,192
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, ...}