Given bounded sequence $\{a_n\}_{n=1}^{\infty}$ which doesn't have minimum or maximum - prove that $a_n$ doesn't converge.
Thanks a lot.
Given bounded sequence $\{a_n\}_{n=1}^{\infty}$ which doesn't have minimum or maximum - prove that $a_n$ doesn't converge.
Thanks a lot.
Hint: $a_n$ is bounded and therefore, it has an infimum and a supremum. Since it does not have a maximum and minimum, it has infinitely many elements near both infimum and supremum. This means that there are 2 subsequences of $a_n$; one of them converges to the infimum and the other converges to the supremum. But infimum and supremum are distinct (if they were not distinct it would be a constant sequence). So, $a_n$ does not converge.
Assume that $a_n \to 0$, just for simplicity. Call $M=\sup_n a_n$ and $m = \inf_n a_n$, so that $M$ and $m$ are finite by assumption, and $m \leq 0 \leq M$. If $M=m$, there is nothing to prove. Otherwise, fix $\epsilon>0$ small, and assume that $m \leq 0 < M$. There exists $N \in \mathbb{N}$ such that $-\epsilon < a_n < \epsilon$ whenever $n > N$. Therefore either $m = \inf \{a_n \mid n \leq N\}$ or $M=\sup \{a_n \mid n \leq N\}$, since $a_n$ can't be arbitrarily close to a number $m \leq 0$ and to a number $M>0$. Hence either $M$ or $m$ is attained.