Terms_of_Bounded_Sequence_Within_Bounds

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