Inertia Principle

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

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

Let $c \in \R$: such that $c < l$.

Then $\exists N \in \N$ such that:
 * $\forall n \in \N: n \ge N \implies c < a_n$

Proof
Pick $\epsilon = l - c > 0$ (as $c < l$).

As $a_n \to l$ as $n \to \infty$, then $\exists N \in \N$ such that:
 * $\forall n \in \N: n \ge N \implies \size {a_n - l} < \epsilon$

Equivalently:
 * $\forall n \in \N: n \ge N \implies \size {l - a_n} < l - c$

For each $a_n$, either $a_n \ge l$ or $a_n < l$.

If $a_n < l$, then $0 < l - a_n$, so $\size {l - a_n} = l - a_n$.

Then:

If $a_n \ge l$, and $l > c$, then $a_n > c$.

So:
 * $\forall n \in \N: n \ge N \implies a_n > c$

Note
Not to be confused with the Principle of Inertia.