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

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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.