196 lines
6.2 KiB
Plaintext
196 lines
6.2 KiB
Plaintext
/-
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Exercises I 3.12
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A Set of Axioms for the Real-Number System
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-/
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import Mathlib.Algebra.Order.Floor
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import Mathlib.Data.PNat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Real.Sqrt
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import Mathlib.Tactic.LibrarySearch
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import Bookshelf.Real.Rational
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import OneVariableCalculus.Apostol.Chapter_I_3
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-- ========================================
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-- Exercise 1
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--
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-- If `x` and `y` are arbitrary real numbers with `x < y`, prove that there is
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-- at least one real `z` satisfying `x < z < y`.
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-- ========================================
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theorem exercise1 (x y : ℝ) (h : x < y) : ∃ z, x < z ∧ z < y := by
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have ⟨z, hz⟩ := exists_pos_add_of_lt' h
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refine ⟨x + z / 2, ⟨?_, ?_⟩⟩
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· have hz' : z / 2 > 0 := by
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have hr := div_lt_div_of_lt (show (0 : ℝ) < 2 by simp) hz.left
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rwa [zero_div] at hr
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exact (lt_add_iff_pos_right x).mpr hz'
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· have hz' : z / 2 < z := div_lt_self hz.left (show 1 < 2 by norm_num)
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calc x + z / 2
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_ < x + z := (add_lt_add_iff_left x).mpr hz'
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_ = y := hz.right
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-- ========================================
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-- Exercise 2
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--
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-- If `x` is an arbitrary real number, prove that there are integers `m` and `n`
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-- such that `m < x < n`.
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-- ========================================
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theorem exercise2 (x : ℝ) : ∃ m n : ℝ, m < x ∧ x < n := by
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refine ⟨x - 1, ⟨x + 1, ⟨?_, ?_⟩⟩⟩ <;> norm_num
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-- ========================================
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-- Exercise 3
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--
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-- If `x > 0`, prove that there is a positive integer `n` such that `1 / n < x`.
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-- ========================================
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theorem exercise3 (x : ℝ) (h : x > 0) : ∃ n : ℕ+, 1 / n < x := by
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have ⟨n, hn⟩ := @Real.exists_pnat_mul_self_geq_of_pos x 1 h
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refine ⟨n, ?_⟩
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have hr := mul_lt_mul_of_pos_right hn (show 0 < 1 / ↑↑n by norm_num)
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conv at hr => arg 2; rw [mul_comm, ← mul_assoc]; simp
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rwa [one_mul] at hr
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-- ========================================
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-- Exercise 4
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--
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-- If `x` is an arbitrary real number, prove that there is exactly one integer
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-- `n` which satisfies the inequalities `n ≤ x < n + 1`. This `n` is called the
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-- greatest integer in `x` and is denoted by `⌊x⌋`. For example, `⌊5⌋ = 5`,
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-- `⌊5 / 2⌋ = 2`, `⌊-8/3⌋ = -3`.
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-- ========================================
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theorem exercise4 (x : ℝ) : ∃! n : ℤ, n ≤ x ∧ x < n + 1 := by
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let n := Int.floor x
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refine ⟨n, ⟨?_, ?_⟩⟩
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· exact ⟨Int.floor_le x, Int.lt_floor_add_one x⟩
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· intro y hy
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rw [← Int.floor_eq_iff] at hy
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exact Eq.symm hy
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-- ========================================
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-- Exercise 5
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--
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-- If `x` is an arbitrary real number, prove that there is exactly one integer
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-- `n` which satisfies `x ≤ n < x + 1`.
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-- ========================================
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theorem exercise5 (x : ℝ) : ∃! n : ℤ, x ≤ n ∧ n < x + 1 := by
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let n := Int.ceil x
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refine ⟨n, ⟨?_, ?_⟩⟩
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· exact ⟨Int.le_ceil x, Int.ceil_lt_add_one x⟩
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· simp only
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intro y hy
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suffices y - 1 < x ∧ x ≤ y by
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rw [← Int.ceil_eq_iff] at this
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exact Eq.symm this
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apply And.intro
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· have := (sub_lt_sub_iff_right 1).mpr hy.right
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rwa [add_sub_cancel] at this
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· exact hy.left
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-- ========================================
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-- Exercise 6
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--
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-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
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-- at least one rational number `r` satisfying `x < r < y`, and hence infinitely
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-- many. This property is often described by saying that the rational numbers
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-- are *dense* in the real-number system.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 7
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--
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-- If `x` is rational, `x ≠ 0`, and `y` irrational, prove that `x + y`, `x - y`,
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-- `xy`, `x / y`, and `y / x` are all irrational.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 8
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--
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-- Is the sum or product of two irrational numbers always irrational?
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 9
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--
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-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
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-- at least one irrational number `z` satisfying `x < z < y`, and hence
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-- infinitely many.
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 10
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--
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-- An integer `n` is called *even* if `n = 2m` for some integer `m`, and *odd*
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-- if `n + 1` is even. Prove the following statements:
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--
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-- (e) Every rational number can be expressed in the form `a / b`, where `a` and
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-- `b` are integers, at least one of which is odd.
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-- ========================================
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def is_even (n : ℤ) := ∃ m : ℤ, n = 2 * m
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def is_odd (n : ℤ) := is_even (n + 1)
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-- ----------------------------------------
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-- (a) An integer cannot be both even and odd.
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- (b) Every integer is either even or odd.
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- (c) The sum or product of two even integers is even. What can you say about
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-- the sum or product of two odd integers?
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-- ----------------------------------------
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-- # TODO
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-- ----------------------------------------
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-- (d) If `n²` is even, so is `n`. If `a² = 2b²`, where `a` and `b` are
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-- integers, then both `a` and `b` are even.
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-- ----------------------------------------
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-- # TODO
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-- ========================================
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-- Exercise 11
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--
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-- Prove that there is no rational number whose square is `2`.
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--
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-- [Hint: Argue by contradiction. Assume `(a / b)² = 2`, where `a` and `b` are
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-- integers, at least one of which is odd. Use parts of Exercise 10 to deduce a
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-- contradiction.]
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-- ========================================
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-- # TODO
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-- ========================================
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-- Exercise 12
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--
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-- The Archimedean property of the real-number system was deduced as a
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-- consequence of the least-upper-bound axiom. Prove that the set of rational
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-- numbers satisfies the Archimedean property but not the least-upper-bound
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-- property. This shows that the Archimedean property does not imply the
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-- least-upper-bound axiom.
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-- ========================================
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-- # TODO
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