Real Analysis: bunch of true and false questions

[squaremeplz]

Problem Statements

a. if a sequence is montone then every one of its subsequences is monotone.

true.

b. if a sequence is not monotone, then every one of its subsequences is not monotone

false.

c. if a sequence is unbounded, then every one of its susequences is unbounded.

true

d. if a sequene is divergent, then it cannot have a convergent subsequence

false

e. if a sequence tends to +inf, then it cannot have a convergent subsequence

true

f. if a sequence is unbounded, then it cannot have a convergent subsequence

false

g. if lim sup s_n = 0, then lim sup|s_n| = 0


false

h. if kim sup|s_n| = 0, then lim sup s_n = 0

true

i. if lim sup|s_n| = 5, then lim sup s_n = 5

true

j. if lim sup|s_n|= 5, then (s_n) is bounded

false


I actually have to prove out the ones that are true but can someone just let me know if I got the first step right on them. Thanks a lot!

 

[HallsofIvy]

Problem Statements

a. if a sequence is montone then every one of its subsequences is monotone.

true.

b. if a sequence is not monotone, then every one of its subsequences is not monotone

false.

c. if a sequence is unbounded, then every one of its susequences is unbounded.

true

What about the sequence 1, 2, 3, 2, 4, 2, 5, 2, 6, 2, ... ?

d. if a sequene is divergent, then it cannot have a convergent subsequence

false

e. if a sequence tends to +inf, then it cannot have a convergent subsequence

true

What about the sequence 1, 2, 3, 2, 4, 2, 5, 2, 6, 2, ... ?

f. if a sequence is unbounded, then it cannot have a convergent subsequence

false

g. if lim sup s_n = 0, then lim sup|s_n| = 0


false

h. if kim sup|s_n| = 0, then lim sup s_n = 0

true

i. if lim sup|s_n| = 5, then lim sup s_n = 5

true

What about the sequence -5, -5, -5, ... ?

j. if lim sup|s_n|= 5, then (s_n) is bounded

false


I actually have to prove out the ones that are true but can someone just let me know if I got the first step right on them. Thanks a lot!

 

[squaremeplz]

thanks so much! those are some attractive sequences.

 

[Office_Shredder]

(e) looks true to me... halls, your counterexample doesn't tend to +infinity

 

[HallsofIvy]

Yes, thanks, Office Shredder, I was thinking of an unbounded sequence.