Inertia Principle

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Theorem

Let $\left \langle {a_n}\right \rangle$ be a sequence in $\R$.

Let $a_n \to l$ as $n \to \infty$, and let $c < l$ where $c \in \R$.


Then $\exists N \in \N$ such that:

$\forall n \in \N, n \ge N: c < a_n$


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 $\forall n \ge N: \left|{a_n - l}\right| < \epsilon$.

Equivalently:

$\forall n \ge N: \left|{l - a_n}\right| < l-c$

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


If $a_n < l$, then $0 < l - a_n$, so $\left|{l - a_n}\right| = l - a_n$.

So $\forall n \ge N: \left|{l - a_n}\right| < l - c \implies l - a_n < l - c \implies a_n > c$.

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

So $\forall n \ge N: a_n > c$.

$\blacksquare$


Note

Not to be confused with the Principle of Inertia.