Write I 3.12 exercise prompts.

one-variable-calculus/Apostol/Chapter_I_3.lean

@@ -1,3 +1,8 @@
+/-
+Chapter I 3
+
+A Set of Axioms for the Real-Number System
+-/
 import Common.Data.Real.Set
 
 #check Archimedean

one-variable-calculus/Apostol/Exercises_I_3_12.lean

@@ -0,0 +1,131 @@
+/-
+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