Inertia Principle
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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:
\(\ds \forall n \in \N: \, \) | \(\ds n\) | \(\ge\) | \(\ds N\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {l - a_n}\) | \(\lt\) | \(\ds l - c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds l - a_n\) | \(\lt\) | \(\ds l - c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_n\) | \(\gt\) | \(\ds c\) |
If $a_n \ge l$, and $l > c$, then $a_n > c$.
So:
- $\forall n \in \N: n \ge N \implies a_n > c$
$\blacksquare$
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Note
Not to be confused with the Principle of Inertia.