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Apr 19, 20251 min read

An alternate definition for Sequence Convergence.

Definition

With sequence ${a_{n}}_{1}^{\infty}$ be a convergent sequence given s.t $a_{n} \to L$ $\forall ϵ > 0, \exists N > 0, \forall n, m \in Z^{+}, n, m > N ⟹ ∣ a_{n} - a_{m} ∣ < ϵ$

A sequence will get closer and closer to itself

Let $ϵ > 0$ be arbitrary
By defn of convergent, $\exists N \in Z^{+}, \forall n \in Z^{+}, n > N ⟹ ∣ a - L ∣ < \frac{ϵ}{4}$
Let $n, m \in Z^{+}$ be aribtrary
Consider $∣ a_{n} - a_{m} ∣ = ∣ a_{n} - a_{m} + L - L ∣$
$= ∣ (a_{n} - L) + (L - a_{m}) ∣$ by alg
$\leq ∣ a_{n} - L ∣ + ∣ L - a_{m} ∣$ by Triangle Inequality
$= ∣ a_{n} - L ∣ + ∣ a_{m} - L ∣$ by $∣ \cdot ∣$ props
$< \frac{ϵ}{4} + \frac{ϵ}{4}$
$= \frac{ϵ}{2} < ϵ$
$□$