MHB Clovis's question at Yahoo Answers (bounded sequence)

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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.
 
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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.
 
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Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
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