Terms of Bounded Sequence Within Bounds

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Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ be bounded.

Let the limit superior of $\sequence {x_n}$ be $\overline l$.

Let the limit inferior of $\sequence {x_n}$ be $\underline l$.


Then:

$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$
$\forall \epsilon > 0: \exists N: \forall n > N: x_n > \underline l - \epsilon$


Proof

Upper Bound

First we show that:

$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$


Aiming for a contradiction, suppose this proposition were to be false.

That would mean that for some $\epsilon > 0$ it would be true that for each $N$ we would be able to find $n > N$ such that $x_n \ge \overline l + \epsilon$.

From Limit of Subsequence of Bounded Sequence it would follow that there exists a convergent subsequence whose limit was $l \ge \overline l + \epsilon$.

This would contradict the definition of $\overline l$.

$\Box$


Lower Bound

Next, in the same way, we show that:

$\forall \epsilon > 0: \exists N: \forall n > N: x_n > \underline l - \epsilon$


Aiming for a contradiction, suppose this proposition were to be false.

That would mean that for some $\epsilon > 0$ it would be true that for each $N$ we would be able to find $n > N$ such that $x_n \le \underline l + \epsilon$.

From Limit of Subsequence of Bounded Sequence it would follow that there exists a convergent subsequence whose limit was $l \le \underline l + \epsilon$.

This would contradict the definition of $\underline l$.

$\blacksquare$


Sources

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