Teoremi Analisi 1

lem succ

∀ε>0 ∃α ∈ ℕ : |αωₙ-ℓ|<ε ∀n ≥ α⁻1

lem pazzo

∀ε>0 ∃x₀>0 : |g(x)-ℓ|<ε ∀x ∈ ]x₀-δ ; x₀+δ[

IPOTESI

lem g(x)=ℓ ⟺ ∀{xₙ}⊆A-{x₀} : xₙ⟶x₀ ⟹ (g(xₙ)-ℓ)⟶ℓ

DIM:

P⇒Q

Q̅⇒P̅

∀ε>0 ∃x₀>0 : |g(x)-ℓ|<ε ∀x ∈ ]x₀-δ ; x₀+δ[

NEGANDO

∃ε>0 : ∀δ>0, |g(x)-ℓ|≥ε per alcuni valori x∈]x₀-δ ; x₀+δ[

δ=1_n

∃{xₙ}⊆]x₀-1_m ; x₀+1_m[ : |g(xₙ)-ℓ|≥ε ∀n ∈ ℕ

x₀-1_m < xₙ < x₀+1_m ⟹ xₙ⟶x₀

Ipotesi

g: A → ℝ

x₀ p.d.c.

Se

lim _x→x₀ g(x) = ℓ

DIM.

lim _x→x₀ g₁ = ℓ

∀ε>0 ∃δ>0 : |g₁₋ℓ|< ε   ∀x∈ (x₀-δ), x≠x₀

ℓ - ε < g(x) < ℓ + ε   ε = ^ℓ⁄₂

ℓ - ^ℓ⁄₂ < g(x) < ℓ + ^ℓ⁄₂

g(x) → ^ℓ⁄₂

Se ℓ = +∞

∀ε>0 ∃δ>0 : g(x)>M   ∀x∈(x₀-δ, +∞) ∉ C

Sia a₁, a₂, a_n...

An monotona -> regolare, cioè ammette limito

Ipotesi

an monotona

crescente => an ≥ a_n+1

-∞ < l < ∞

an → l ∈ ℝ∞

1. an crescente, limitata
2. an crescente, ma illimitata

Dim. 1

Ris. per assur. poiché monot.

∀ɛ > 0 ∃n̄ : l - ɛ < a_n∞

• l - ɛ < a_n ∞ an ≤ l ≤ l + ɛ
• l - ɛ con α ɛ < ∊

an ≠ l

an < an crescente

∃ n̄ : (an_n̄) ∞ ≥ M

an > an_n ∞ ≥ M

an ≥ M

an > an_n

falso, ∃ n̄ : (an_n̄) ≥ M ∀n ≥ n̄

an ∞ ≥ an ∞ ≥ M