Archimedean property and consistent theorem environment.

common/Common/Sequence/Arithmetic.tex

@@ -3,6 +3,11 @@
 \usepackage{hyperref}
 
 \newtheorem{theorem}{Theorem}
+\newtheorem{custominner}{Theorem}
+\newenvironment{custom}[1]{%
+  \renewcommand\thecustominner{#1}%
+  \custominner
+}{\endcustominner}
 
 \begin{document}
 

common/Common/Sequence/Geometric.tex

@@ -3,6 +3,11 @@
 \usepackage{hyperref}
 
 \newtheorem{theorem}{Theorem}
+\newtheorem{custominner}{Theorem}
+\newenvironment{custom}[1]{%
+  \renewcommand\thecustominner{#1}%
+  \custominner
+}{\endcustominner}
 
 \begin{document}
 

mathematical-introduction-logic/Enderton/Chapter0.tex

@@ -3,6 +3,11 @@
 \usepackage{hyperref}
 
 \newtheorem{theorem}{Theorem}
+\newtheorem{custominner}{Theorem}
+\newenvironment{custom}[1]{%
+  \renewcommand\thecustominner{#1}%
+  \custominner
+}{\endcustominner}
 
 \begin{document}
 

one-variable-calculus/Apostol.lean

@@ -1 +1 @@
-def hello := "world"
+import Apostol.Chapter_I_3_10

one-variable-calculus/Apostol/Chapter_I_3_10.lean

@@ -0,0 +1,60 @@
+import Mathlib.Data.PNat.Basic
+import Mathlib.Data.Real.Basic
+import Mathlib.Order.Basic
+import Mathlib.Tactic.LibrarySearch
+
+#check Archimedean
+
+namespace Real
+
+/--
+Every real should be less than or equal to the absolute value of its ceiling.
+-/
+lemma leq_nat_abs_ceil_self (x : ℝ) : x ≤ Int.natAbs ⌈x⌉ := by
+  by_cases h : x ≥ 0
+  · let k : ℤ := ⌈x⌉
+    unfold Int.natAbs
+    have k' : k = ⌈x⌉ := rfl
+    rw [←k']
+    have _ : k ≥ 0 := by  -- Hint for match below
+      rw [k', ge_iff_le]
+      exact Int.ceil_nonneg (ge_iff_le.mp h)
+    match k with
+    | Int.ofNat m => calc x
+      _ ≤ ⌈x⌉ := Int.le_ceil x
+      _ = Int.ofNat m := by rw [←k']
+  · have h' : ((Int.natAbs ⌈x⌉) : ℝ) ≥ 0 := by simp
+    calc x
+      _ ≤ 0 := le_of_lt (lt_of_not_le h)
+      _ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'    
+
+/--
+Theorem 1.29
+
+For every real `x` there exists a positive integer `n` such that `n > x`.
+-/
+theorem exists_pnat_geq_self (x : ℝ) : ∃ n : ℕ+, ↑n > x := by
+  let x' : ℕ+ := ⟨Int.natAbs ⌈x⌉ + 1, by simp⟩
+  have h : x < x' := calc x
+    _ ≤ Int.natAbs ⌈x⌉ := leq_nat_abs_ceil_self x
+    _ < ↑↑(Int.natAbs ⌈x⌉ + 1) := by simp
+    _ = x' := rfl
+  exact ⟨x', h⟩
+
+/--
+Theorem 1.30
+
+If `x > 0` and if `y` is an arbitrary real number, there exists a positive
+integer `n` such that `nx > y`.
+
+This is known as the *Archimedean Property of the Reals*.
+-/
+theorem pos_imp_exists_pnat_mul_self_geq {x y : ℝ}
+  : x > 0 → ∃ n : ℕ+, n * x > y := by
+  intro hx
+  let ⟨n, p⟩ := exists_pnat_geq_self (y / x)
+  have p' := mul_lt_mul_of_pos_right p hx
+  rw [div_mul, div_self (show x ≠ 0 from LT.lt.ne' hx), div_one] at p'
+  exact ⟨n, p'⟩
+
+end Real

one-variable-calculus/Apostol/Chapter_I_3_10.tex

@@ -0,0 +1,38 @@
+\documentclass{article}
+\usepackage{amsfonts, amsthm}
+\usepackage{hyperref}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{custominner}{Theorem}
+\newenvironment{custom}[1]{%
+  \renewcommand\thecustominner{#1}%
+  \custominner
+}{\endcustominner}
+
+\begin{document}
+
+\begin{custom}{1.29}
+
+For every real $x$ there exists a positive integer $n$ such that $n > x$.
+
+\end{custom}
+
+\begin{proof}
+
+\href{Chapter_I_3_10.lean}{Apostol.Chapter\_I\_3\_10.Real.exists\_pnat\_geq\_self}
+
+\end{proof}
+
+\begin{custom}{1.30}[Archimedean Property of the Reals]
+
+If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$.
+
+\end{custom}
+
+\begin{proof}
+
+\href{Chapter_I_3_10.lean}{Apostol.Chapter\_I\_3\_10.Real.pos\_imp\_exists\_pnat\_mul\_self\_geq}
+
+\end{proof}
+
+\end{document}

one-variable-calculus/README.md

@@ -0,0 +1,4 @@
+# One-Variable Calculus, with an Introduction to Linear Algebra
+
+Apostol, Tom M. Calculus, Vol. 1: One-Variable Calculus, with an Introduction to
+Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991.