Proving the Maximal Member in a Converging Sequence An > 0 to 0

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In summary, a bounded sequence that converges to 0 will have a maximal member, which is the first member of the sequence. This can be proven mathematically by combining the fact that the sequence is descending and converges to 0, and the fact that finitely many members of the sequence take on a different value while the rest approach 0.
  • #1
transgalactic
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An is a bounded sequence which converges to 0.
An>0 for every n.
prove that in this sequence we have a maximal member
??

again its obvious because if a sequence is descending and it converges from a positive number
to 0.
so 0 is the larger lower bound
and our first member must be the larger

how to transform it to math?
 
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  • #2
If it converges to 0, then finitely many members of the sequence take on a different value (that number may be small or large but it is still finite) and the rest of the terms approach 0.
 
  • #3
how to prove it mathematically?
 
  • #4
By combining what you and I wrote and using proper notation
 

1. What is a converging sequence?

A converging sequence is a sequence of numbers that gets closer and closer to a specific value as the sequence progresses.

2. What does it mean for a sequence to have a maximal member?

A maximal member in a sequence is the largest number in the sequence.

3. Why is it important to prove the maximal member in a converging sequence?

Proving the maximal member in a converging sequence is important because it can help us understand the behavior of the sequence and can be used to make predictions about the sequence's future values.

4. How do you prove the maximal member in a converging sequence?

To prove the maximal member in a converging sequence An > 0 to 0, you can use the limit comparison test or the ratio test. Both of these tests involve comparing the sequence to a known converging sequence and using the comparison to determine the behavior of the sequence.

5. Can a sequence have more than one maximal member?

No, a sequence can only have one maximal member. This is because the maximal member is defined as the largest number in the sequence, so there can only be one.

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