Clovis's question at Yahoo Answers (bounded sequence)

[Fernando Revilla]

Here is the question:

Construct a bounded sequence that has no maximum and no minimum element. Could a sequence like this converge? Why or why not?

Here is a link to the question:

Real Analysis Sequences 10pts? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Clovis,

Consider the sequence $a_n=\dfrac{(-1)^nn}{n+1}$. Easily proved, $-1\leq a_n\leq 1$ for all $n$ and the sequence has no maximum and no minimum element.

Now, suppose $x_n$ is bounded sequence, then it has a infimum $m$ and a supremum $M$. If $x_n$ has no maximum and minimum, it has infinitely many elements close both infimum and supremum, so we can construct two subsequences $x_{n_k}\to m$ and $x_{n_r}\to M$. But $m\neq M$ (otherwise, $x_n$ would be a constant sequence, i.e. with maximum and minimum). This means that $x_n$ does not converge.