Lower and Upper Bounds for Sequences/Warning

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

Let $x_n \to l$ as $n \to \infty$.

Then it is not necessarily the case that:


 * $(1): \quad \forall n \in \N: x_n > a \implies l > a$
 * $(2): \quad \forall n \in \N: x_n < b \implies l < b$

Proof
Take the examples:


 * $(1): \quad \sequence {x_n} = \dfrac 1 n$
 * $(2): \quad \sequence {y_n} = -\dfrac 1 n$

Then :
 * $\forall n \in \N_{>0}: \dfrac 1 n > 0, -\dfrac 1 n < 0$

From Sequence of Reciprocals is Null Sequence, we have
 * $x_n \to 0$
 * $y_n \to 0$

as $n \to \infty$.

However, it is clearly false that $0 > 0$ and $0 < 0$.