Calculemus (Posts about LeanProver)

Challenge 23: The constant sequence aₙ = L converges to L

José A. Alonso — Mon, 12 Oct 2026 04:00:00 GMT

Prove that the constant sequence \(a_{n} = L\) converges to \(L\).

To do this, complete the following theory:

import Mathlib.Tactic

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

variable {a : ℕ → ℝ}
variable {L : ℝ}

example
  (h : ∀ n, a n = L)
  : SeqLim a L :=
by sorry

How to publish your solution?

Write your proof in live.lean-lang.org and copy the generated link.
Post it on the Mastodon thread for this problem, with a message like:
"My solution is at [link]."
You will be able to compare your approach with other participants during the week. The official solution will be published next Monday.

Lean 4 for mathematicians: a teacher's guide

José A. Alonso — Sat, 10 Oct 2026 08:00:00 GMT

Introduction

Who it is for. This guide is aimed at teachers who want to introduce Lean 4 to mathematics students who are used to writing proofs on paper but have no previous programming experience. No prior knowledge of computer science is required.

Philosophy of the course.

Lean is a proof notebook that talks back. You write an argument and the computer tells you, step by step, what is missing. It is presented as a patient interlocutor, not as a programming language.
Mathematics first, syntax later. Each topic starts from a proof the student already knows how to do on paper and translates it into Lean.
No installation at the beginning. The first three sessions can be done in the browser. Local installation comes when the student already wants to carry on alone.
You learn by reading the proof state. The central skill is not memorizing tactics, but reading the goals window (the infoview) the way one reads a blackboard.
Making mistakes is cheap. The red error is information, not a failure.

Structure. Eight topics of about 2 hours each (theory and practice mixed). Each topic has two parts: the content (what the student sees) and the teaching tips (notes for the teacher with key ideas, typical mistakes, ...).

Convention. All the examples start with import Mathlib.Tactic, Lean's library of tactics. The code blocks can be pasted as they are into the editor.

Topic	Title	Central idea
1	First contact	A theorem is a type; a proof is a term
2	Propositional logic	Connectives and the tactics that handle them
3	Quantifiers and sets	∀, ∃, membership, inclusion
4	Equality and calculation	`rw`, `calc`, `ring`, `linarith`
5	Induction and recursion	Natural numbers, sums, recursive definitions
6	Your own definitions	`def`, `structure`, limits
7	Finding your way around Mathlib	Looking up lemmas without knowing them by heart
8	Final project	Formalizing a result of your own

Challenge 22: Subsequences with different limits imply non-convergence

José A. Alonso — Mon, 05 Oct 2026 04:00:00 GMT

Prove that if a sequence has two subsequences with different limits, then the sequence is not convergent.

To do so, complete the following theory:

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

-- a converges to L
def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop :=
  ∀ ε > 0, ∃ k : ℕ, ∀ n ≥ k, |a n - L| < ε

-- The sequence a is convergent
def SeqConv (a : ℕ → ℝ) : Prop :=
  ∃ L, SeqLim a L

-- φ is an extraction function.
def IsExtraction (φ : ℕ → ℕ) : Prop :=
  StrictMono φ

-- b is a subsequence of a.
def Subseq (b a : ℕ → ℝ) : Prop :=
  ∃ φ, IsExtraction φ ∧ b = a ∘ φ

variable {a b₁ b₂ : ℕ → ℝ}
variable {L L₁ L₂ : ℝ}

example
  (hb₁ : Subseq b₁ a)
  (hL₁ : SeqLim b₁ L₁)
  (hb₂ : Subseq b₂ a)
  (hL₂ : SeqLim b₂ L₂)
  (h : L₁ ≠ L₂)
  : ¬SeqConv a :=
by sorry

Challenge 21: Subsequences have the same limit as the sequence

José A. Alonso — Mon, 28 Sep 2026 04:00:00 GMT

A subsequence is obtained by applying an extraction function to the original sequence; that is, a strictly increasing function \(φ : ℕ \to ℕ\). For example, the subsequence \[ u_{0}, u_{2}, u_{4}, u_{6}, \dots \] is obtained with the extraction function \(\varphi\) defined by \(\varphi(n) = 2n\).

The previous definitions are formalized in Lean 4 as:

-- φ is an extraction function.
def IsExtraction (φ : ℕ → ℕ) :=
  StrictMono φ

-- v is a subsequence of u.
def Subseq (v u : ℕ → ℝ) :=
  ∃ φ, IsExtraction φ ∧ v = u ∘ φ

-- a is the limit of u.
def SeqLim (u : ℕ → ℝ) (a : ℝ) :=
  ∀ ε > 0, ∃ k : ℕ, ∀ n ≥ k, |u n - a| < ε

Prove that every subsequence of a convergent sequence converges to the same limit as the sequence.