Calculemus (Publicaciones sobre Limits)

If uₙ tends to a y vₙ tends to b, then uₙvₙ tends to ab

José A. Alonso — Mon, 02 Dec 2024 04:00:00 GMT

In Lean4, a sequence \(u_0, u_1, u_2, ...\) can be represented by a function \(u : ℕ → ℝ\) such that \(u(n)\) is \(u_n\).

It is defined that the sequence \(u\) tends to \(c\) by:

   def TendsTo (u : ℕ → ℝ) (c : ℝ) :=
     ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - c| < ε

Prove that if \(u_n\) tends to \(a\) and \(v_n\) tends to \(b\), then \(u_nv_n\) tends to \(ab\).

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic
import Mathlib.Tactic

variable {u v : ℕ → ℝ}
variable {a b : ℝ}

def TendsTo (u : ℕ → ℝ) (c : ℝ) :=
  ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - c| < ε

example
  (hu : TendsTo u a)
  (hv : TendsTo v b)
  : TendsTo (fun n ↦ u n * v n) (a * b) :=
by sorry

Leer más… (quedan 12 minutos de lectura)

If the limit of the sequence u(n) is a and c ∈ ℝ, then the limit of u(n)c is ac

José A. Alonso — Fri, 29 Nov 2024 04:00:00 GMT

In Lean, a sequence \(u_₀, u_₁, u_₂, ... \) can be represented by a function \(u : ℕ → ℝ\) such that \(u(n)\) is \(u_n\).

It is defined that \(a\) is the limit of the sequence \(u\), by

   def limite : (ℕ → ℝ) → ℝ → Prop :=
     fun u c ↦ ∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - c| < ε

Prove that if the limit of \(u_n\) is \(a\), then the limit of \(u_nc\) is \(ac\).

import Mathlib.Data.Real.Basic
import Mathlib.Tactic

variable (u : ℕ → ℝ)
variable (a c : ℝ)

def TendsTo : (ℕ → ℝ) → ℝ → Prop :=
  fun u c ↦ ∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - c| < ε

example
  (h : TendsTo u a)
  : TendsTo (fun n ↦ (u n) * c) (a * c) :=
by sorry

Leer más… (quedan 4 minutos de lectura)

If u(n) tends to a, then 7u(n) tends to 7a

José A. Alonso — Sun, 10 Nov 2024 04:00:00 GMT

In Lean, a sequence \(u_0, u_1, u_2, ...\) can be represented by a function \(u : ℕ → ℝ\) such that \(u(n)\) is the term \(u_n\).

We define that \(u\) tends to \(a\) by

   def tendsTo : (ℕ → ℝ) → ℝ → Prop :=
     fun u c ↦ ∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - c| < ε

Prove that if \(u_n\) tends to \(a\), then \(7u\_n\) tends to \(7a\).

To do this, complete the following Lean4 theory:

import Mathlib.Tactic

variable {u : ℕ → ℝ}
variable {a : ℝ}

def tendsTo : (ℕ → ℝ) → ℝ → Prop :=
  fun u c ↦ ∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - c| < ε

example
  (h : tendsTo u a)
  : tendsTo (fun n ↦ 7 * u n) (7 * a) :=
by sorry

Leer más… (quedan 4 minutos de lectura)

If f is continuous at a and the limit of u(n) is a, then the limit of f(u(n)) is f(a)

José A. Alonso — Wed, 04 Sep 2024 04:00:00 GMT

In Lean4, we can define that \(a\) is the limit of the sequence \(u\) by:

   def is_limit (u : ℕ → ℝ) (a : ℝ) :=
     ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| < ε

and that \(f\) is continuous at \(a\) by:

   def continuous_at (f : ℝ → ℝ) (a : ℝ) :=
     ∀ ε > 0, ∃ δ > 0, ∀ x, |x - a| ≤ δ → |f x - f a| ≤ ε

To prove that if the function \(f\) is continuous at the point \(a\), and the sequence \(u(n)\) converges to \(a\), then the sequence \(f(u(n))\) converges to \(f(a)\).

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic

variable {f : ℝ → ℝ}
variable {a : ℝ}
variable {u : ℕ → ℝ}

def is_limit (u : ℕ → ℝ) (a : ℝ) :=
  ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| ≤ ε

def continuous_at (f : ℝ → ℝ) (a : ℝ) :=
  ∀ ε > 0, ∃ δ > 0, ∀ x, |x - a| ≤ δ → |f x - f a| ≤ ε

example
  (hf : continuous_at f a)
  (hu : is_limit u a)
  : is_limit (f ∘ u) (f a) :=
by sorry

Leer más… (quedan 7 minutos de lectura)

If x is the limit of u and y is an upper bound of u, then x ≤ y

José A. Alonso — Tue, 03 Sep 2024 04:00:00 GMT

In Lean4, we can define that \(a\) is the limit of the sequence \(u\) by:

   def is_limit (u : ℕ → ℝ) (a : ℝ) :=
     ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| < ε

and that \(a\) is an upper bound of \(u\) by:

   def is_upper_bound (u : ℕ → ℝ) (a : ℝ) :=
     ∀ n, u n ≤ a

Prove that if \(x\) is the limit of the sequence \(u\) and \(y\) is an upper bound of \(u\), then \(x ≤ y\).

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic
import Mathlib.Tactic

variable  (u : ℕ → ℝ)
variable (x y : ℝ)

def is_limit (u : ℕ → ℝ) (a : ℝ) :=
  ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| < ε

def is_upper_bound (u : ℕ → ℝ) (a : ℝ) :=
  ∀ n, u n ≤ a

example
  (hx : is_limit u x)
  (hy : is_upper_bound u y)
  : x ≤ y :=
by sorry

Leer más… (quedan 5 minutos de lectura)