about sequenses?

Question

Define the supremum sup A of abounded
subset A of R......""i will use x-n in sted of x sub n....""
let (x-n)be bounded sequence.define the sequence S-n by S-n = sup{x-n:k>=n} for n belonges to natural numbers "N".prove that the sequence S-n is converge...
i hope that the prove will be in details
thanx

i mean prove that S-nn is convergent

i mean prove that S-n is convergent

Answer 1

maybe u should take a walk and think about your life!!

Answer 2

I think you've garbled your n's and k's. but here's the general idea (you can fill in the details yourself, since it IS your homework):

1) show that the subsequence in question in bounded (that's easy)

2) show that it is monotone (this will follow if you clean up the n's and k's in the definition.) A good way to begin to think about this is to take examples of specific bounded non-convergent sequences, and list the members of the subsequence in order....I think you'll see what's going on. But of course the examples do not make a proof.

A bounded monotone sequence must converge.

Answer 3

i don't really understand your question. You have x-n is a bounded sequence. S-n = sup{x-n; k>=n}. But if S-n is equal to this, then it is just the smallest upper bound of the sequence x-n. This has only one smallest upper bound. So the sequence S-n just contains one value? which obviously converges to that value - am i missing something?

Answer 4

2 points

Answer 5

e=mc2