# More sequences

#### Carla1985

##### Member
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.

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
Looks good to me.

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

Last edited:

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar

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
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, ...}