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Matt B.
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Matt B. said:Homework Statement
: [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations
: [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution
:[/B] Given a = sup S. We know that a = sup S if: 1) a ∈ S and a is called an upper bound, and 2) if b is also an upper bound, then b ≥ a. Since a = sup S, given ε>0 and the xn ∈ S, we know that a - ε < xn ≤ a since a is a least upper bound. This means that since xn ∈ S and a = sup S, that xn can never exceed the value of a, given as sup S.** I am stuck, any help is beneficial.
My interpretation of the OP was that S is a set (ie unordered) not a sequence. There is a sequence in the set [tex] S = \{1, 1/2, 1/4, 1/8, 1/16, \ldots \} [/tex] that converges to 1, which is the sequence ##x_n=1\forall n##..Ray Vickson said:You will have trouble proving this, because it is false. Here is a simple counterexample:
[tex] S = \{1, 1/2, 1/4, 1/8, 1/16, \ldots \} [/tex]
We have ##a = \sup S = 1##, but there is no subsequence of ##S## that converges to 1.
andrewkirk said:My interpretation of the OP was that S is a set (ie unordered) not a sequence. There is a sequence in the set [tex] S = \{1, 1/2, 1/4, 1/8, 1/16, \ldots \} [/tex] that converges to 1, which is the sequence ##x_n=1\forall n##..
The supremum of a sequence is the least upper bound, or the smallest number that is greater than or equal to all the elements in the sequence.
In order to prove that a sequence converges to its supremum, you need to show that the sequence is increasing and bounded above by the supremum. This can be done using the definition of supremum and the properties of sequences.
No, a sequence can only have one limit that is equal to the supremum. This is because the supremum is the smallest number that is greater than or equal to all the elements in the sequence, so any other limit would be smaller and therefore not the supremum.
Yes, it is possible for a sequence to have a limit that is not equal to the supremum. This can happen if the sequence is not increasing or if it is not bounded above by the supremum.
Showing that a sequence converges to its supremum is a specific case of proving convergence. It demonstrates that the sequence is approaching a specific limit, which in this case is the supremum. This is important in analysis and other fields of mathematics where convergence is a fundamental concept.