Question

True/False Counterexample Help

1. Every oscillating sequence has a convergent sequence.

2. Every oscillating sequence diverges.

Answer 1

I believe that 1 is true but 2 is false.

Answer 2

no idea
what is the definition of an oscillating sequence?

Answer 3

\((-1)^{n}\)??

Answer 4

i assume the first one reads
"Every oscillating sequence has a convergent SUBsequence"

Answer 5

The book definition says that if
lim inf sn < lim sup sn then it is said that sn oscillates.

Answer 6

and yes that is what it says.

Answer 7

but to me, a sequence that goes like -4, 4, -4, 4, -4.... I thought that is what oscillating means

Answer 8

But for 1 if i say that sn = (-1)^n/n.... that would work as a counter example right?? or no

Answer 9

i guess maybe not
it doesn't have to alternate

Answer 10

i mean for 2..

Answer 11

Okay i guess Im confused on the difference between oscillate and alternate.

Answer 12

i would say no because in your example both the lim inf and lim sup are zero

Answer 13

@tkhunny may correct me, but as i recall the limit only exists if the lim sup is the same as the lim inf

Answer 14

It just means that it never settles down. It can do anything between those two limits. lim sup and lim inf. If lim sup and lim inf are the same, the sequence eventually has to hold still!

Answer 15

So then is it that both of them are true??