🍈 Zettelkasten

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Apr 19, 20251 min read

Theorem

Suppose ${a_{n}}$ converges
Choose arbitrary $l_{1}, l_{2}$ s.t ${a_{n}} = l_{1}$ and ${a_{n}} = l_{2}$
We want to show that $l_{1} = l_{2}$
1. $⟹ l_{1} - l_{2} = 0$
2. $⟹ \forall ϵ > 0, ∣ l_{1} - l_{2} ∣ < ϵ$
Let $ϵ > 0$ be arbitrary
Invoke 1 with $ϵ_{1} = \frac{ϵ}{2}$
Note that $\exists N > 0$ s.t if $n > N$ then $∣ a_{n} - l_{1} ∣ < ϵ_{1}$
Invoke 1 with $ϵ_{2} = \frac{ϵ}{2}$
Note that $\exists N > 0$ s.t if $n > N$ then $∣ a_{n} - l_{2} ∣ < ϵ_{2}$
Then, consider $∣ l_{1} - l_{2} ∣ = ∣ - a_{n} - l_{1} + a_{n} - l_{n} ∣ \leq ∣ a_{n} - l_{1} ∣ + ∣ a_{n} - l_{2} ∣ = \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ$
$□$