bookshelf/one-variable-calculus/Apostol/Exercises/Exercises_I_3_12.lean

/-
I 3.12 Exercises

A Set of Axioms for the Real-Number System
-/

-- ========================================
-- Exercise 1
--
-- If `x` and `y` are arbitrary real numbers with `x < y`, prove that there is
-- at least one real `z` satisfying `x < z < y`.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 2
--
-- If `x` is an arbitrary real number, prove that there are integers `m` and `n`
-- such that `m < x < n`.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 3
--
-- If `x > 0`, prove that there is a positive integer `n` such that `1 / n < x`.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 4
--
-- If `x` is an arbitrary real number, prove that there is exactly one integer
-- `n` which satisfies the inequalities `n ≤ x < n + 1`. This `n` is called the
-- greatest integer in `x` and is denoted by `⌊x⌋`. For example, `⌊5⌋ = 5`,
-- `⌊5 / 2⌋ = 2`, `⌊-8/3⌋ = -3`.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 5
--
-- If `x` is an arbitrary real number, prove that there is exactly one integer
-- `n` which satisfies `x ≤ n < x + 1`.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 6
--
-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
-- at least one rational number `r` satisfying `x < r < y`, and hence infinitely
-- many. This property is often described by saying that the rational numbers
-- are *dense* in the real-number system.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 7
--
-- If `x` is rational, `x ≠ 0`, and `y` irrational, prove that `x + y`, `x - y`,
-- `xy`, `x / y`, and `y / x` are all irrational.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 8
--
-- Is the sum or product of two irrational numbers always irrational?
-- ========================================

-- # TODO

-- ========================================
-- Exercise 9
--
-- If `x` and `y` are arbitrary real numbers, `x < y`, prove that there exists
-- at least one irrational number `z` satisfying `x < z < y`, and hence
-- infinitely many.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 10
--
-- An integer `n` is called *even* if `n = 2m` for some integer `m`, and *odd*
-- if `n + 1` is even. Prove the following statements:
--
-- (a) An integer cannot be both even and odd.
-- (b) Every integer is either even or odd.
-- (c) The sum or product of two even integers is even. What can you say about
-- the sum or product of two odd integers?
-- (d) If `n²` is even, so is `n`. If `a² = 2b²`, where `a` and `b` are
-- integers, then both `a` and `b` are even.
-- (e) Every rational number can be expressed in the form `a / b`, where `a` and
-- `b` are integers, at least one of which is odd.
-- ========================================

-- # TODO

-- ========================================
-- Exercise 11
--
-- Prove that there is no rational number whose square is `2`.
--
-- [Hint: Argue by contradiction. Assume `(a / b)² = 2`, where `a` and `b` are
-- integers, at least one of which is odd. Use parts of Exercise 10 to deduce a
-- contradiction.]
-- ========================================

-- # TODO

-- ========================================
-- Exercise 12
--
-- The Archimedean property of the real-number system was deduced as a
-- consequence of the least-upper-bound axiom. Prove that the set of rational
-- numbers satisfies the Archimedean property but not he least-upper-bound
-- property. This shows that the Archimedean property does not imply the
-- least-upper-bound axiom.
-- ========================================

-- # TODO