Sequence Limit

Definition: $latex text{Sequence } (a_n) $
has limit a

$latex boxed{forall varepsilon >0, exists N, forall n geq N text { such that } |(a_n) -a| < varepsilon}$

$latex Updownarrow $

$latex displaystyle boxed{ lim_{ntoinfty} (a_n) = a }$

What if we reverse the order of the definition like this:

∃ N such that ∀ε > 0, ∀n ≥ N,
$latex |(a_n) -a| < varepsilon$

This means:

$latex boxed {forall n geq N, (a_n) = a }$

Example:

$latex displaystyle (a_n) = frac{3n^{2} + 2n +1}{n^{2}-n-3}$

$latex displaystyletext{Prove: } (a_n) text { convergent? If so, what is the limit ?}$

Proof:
$latex displaystyle (a_n) = 3 + frac{5n +10}{n^{2}-n-3}$

$latex n to infty, (a_n) to 3$

Let’s prove it.

$latex text {Let } varepsilon >0$
$latex text{Choose N such that } forall n geq N, $
$latex displaystyle |(a_n) -3| = Bigr|frac{5n +10}{n^{2}-n-3}Bigr| < varepsilon$

$latex text{Simplify: }…